Practice · Same exponent, so match the bases · Sensim Math

Variant 1 easy

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when both its base and its exponent are multiplied by $2$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with both the base and the exponent multiplied by $2$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base and its exponent each multiplied by $2$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

2

gives

2a

; multiplying the exponent by

2

gives

2b

.

S=(2a)^{2b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

2b

as

2\times b

and use the power-of-a-power rule, then raise

2a

to the power

2

.

(2a)^{2b}=\{(2a)^2\}^b=(4a^2)^b

(2a)^2=2^2a^2=4a^2

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(4a^2)^b=(ax)^b \Rightarrow 4a^2=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{4a^2}{a}=4a

a^2\div a=a^{2-1}=a

.

Answer:

x=4a

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=4^2=16$ , and $x=4\times2=8$ gives $a^b\times x^b=2\times8=16$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(4a^2)^b}{a^b}=\left(\frac{4a^2}{a}\right)^b=(4a)^b$ , and since $x>0$ , $x=4a$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 2 easy

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when both its base and its exponent are multiplied by $3$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with both the base and the exponent multiplied by $3$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base and its exponent each multiplied by $3$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

3

gives

3a

; multiplying the exponent by

3

gives

3b

.

S=(3a)^{3b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

3b

as

3\times b

and use the power-of-a-power rule, then raise

3a

to the power

3

.

(3a)^{3b}=\{(3a)^3\}^b=(27a^3)^b

(3a)^3=3^3a^3=27a^3

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(27a^3)^b=(ax)^b \Rightarrow 27a^3=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{27a^3}{a}=27a^2

a^3\div a=a^{3-1}=a^2

.

Answer:

x=27a^2

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=6^3=216$ , and $x=27\times2^2=108$ gives $a^b\times x^b=2\times108=216$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(27a^3)^b}{a^b}=\left(\frac{27a^3}{a}\right)^b=(27a^2)^b$ , and since $x>0$ , $x=27a^2$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 3 easy

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $3$ and its exponent is multiplied by $2$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $3$ and the exponent multiplied by $2$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $3$ and its exponent multiplied by $2$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

3

gives

3a

; multiplying the exponent by

2

gives

2b

.

S=(3a)^{2b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

2b

as

2\times b

and use the power-of-a-power rule, then raise

3a

to the power

2

.

(3a)^{2b}=\{(3a)^2\}^b=(9a^2)^b

(3a)^2=3^2a^2=9a^2

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(9a^2)^b=(ax)^b \Rightarrow 9a^2=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{9a^2}{a}=9a

a^2\div a=a^{2-1}=a

.

Answer:

x=9a

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=6^2=36$ , and $x=9\times2=18$ gives $a^b\times x^b=2\times18=36$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(9a^2)^b}{a^b}=\left(\frac{9a^2}{a}\right)^b=(9a)^b$ , and since $x>0$ , $x=9a$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 4 easy

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $2$ and its exponent is multiplied by $3$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $2$ and the exponent multiplied by $3$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $2$ and its exponent multiplied by $3$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

2

gives

2a

; multiplying the exponent by

3

gives

3b

.

S=(2a)^{3b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

3b

as

3\times b

and use the power-of-a-power rule, then raise

2a

to the power

3

.

(2a)^{3b}=\{(2a)^3\}^b=(8a^3)^b

(2a)^3=2^3a^3=8a^3

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(8a^3)^b=(ax)^b \Rightarrow 8a^3=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{8a^3}{a}=8a^2

a^3\div a=a^{3-1}=a^2

.

Answer:

x=8a^2

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=4^3=64$ , and $x=8\times2^2=32$ gives $a^b\times x^b=2\times32=64$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(8a^3)^b}{a^b}=\left(\frac{8a^3}{a}\right)^b=(8a^2)^b$ , and since $x>0$ , $x=8a^2$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 5 medium

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $2$ and its exponent is multiplied by $4$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $2$ and the exponent multiplied by $4$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $2$ and its exponent multiplied by $4$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

2

gives

2a

; multiplying the exponent by

4

gives

4b

.

S=(2a)^{4b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

4b

as

4\times b

and use the power-of-a-power rule, then raise

2a

to the power

4

.

(2a)^{4b}=\{(2a)^4\}^b=(16a^4)^b

(2a)^4=2^4a^4=16a^4

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(16a^4)^b=(ax)^b \Rightarrow 16a^4=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{16a^4}{a}=16a^3

a^4\div a=a^{4-1}=a^3

.

Answer:

x=16a^3

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=4^4=256$ , and $x=16\times2^3=128$ gives $a^b\times x^b=2\times128=256$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(16a^4)^b}{a^b}=\left(\frac{16a^4}{a}\right)^b=(16a^3)^b$ , and since $x>0$ , $x=16a^3$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 6 medium

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $4$ and its exponent is multiplied by $3$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $4$ and the exponent multiplied by $3$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $4$ and its exponent multiplied by $3$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

4

gives

4a

; multiplying the exponent by

3

gives

3b

.

S=(4a)^{3b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

3b

as

3\times b

and use the power-of-a-power rule, then raise

4a

to the power

3

.

(4a)^{3b}=\{(4a)^3\}^b=(64a^3)^b

(4a)^3=4^3a^3=64a^3

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(64a^3)^b=(ax)^b \Rightarrow 64a^3=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{64a^3}{a}=64a^2

a^3\div a=a^{3-1}=a^2

.

Answer:

x=64a^2

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=8^3=512$ , and $x=64\times2^2=256$ gives $a^b\times x^b=2\times256=512$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(64a^3)^b}{a^b}=\left(\frac{64a^3}{a}\right)^b=(64a^2)^b$ , and since $x>0$ , $x=64a^2$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 7 medium

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $4$ and its exponent is multiplied by $2$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $4$ and the exponent multiplied by $2$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $4$ and its exponent multiplied by $2$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

4

gives

4a

; multiplying the exponent by

2

gives

2b

.

S=(4a)^{2b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

2b

as

2\times b

and use the power-of-a-power rule, then raise

4a

to the power

2

.

(4a)^{2b}=\{(4a)^2\}^b=(16a^2)^b

(4a)^2=4^2a^2=16a^2

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(16a^2)^b=(ax)^b \Rightarrow 16a^2=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{16a^2}{a}=16a

a^2\div a=a^{2-1}=a

.

Answer:

x=16a

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=8^2=64$ , and $x=16\times2=32$ gives $a^b\times x^b=2\times32=64$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(16a^2)^b}{a^b}=\left(\frac{16a^2}{a}\right)^b=(16a)^b$ , and since $x>0$ , $x=16a$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 8 medium

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $3$ and its exponent is multiplied by $4$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $3$ and the exponent multiplied by $4$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $3$ and its exponent multiplied by $4$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

3

gives

3a

; multiplying the exponent by

4

gives

4b

.

S=(3a)^{4b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

4b

as

4\times b

and use the power-of-a-power rule, then raise

3a

to the power

4

.

(3a)^{4b}=\{(3a)^4\}^b=(81a^4)^b

(3a)^4=3^4a^4=81a^4

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(81a^4)^b=(ax)^b \Rightarrow 81a^4=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{81a^4}{a}=81a^3

a^4\div a=a^{4-1}=a^3

.

Answer:

x=81a^3

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=6^4=1296$ , and $x=81\times2^3=648$ gives $a^b\times x^b=2\times648=1296$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(81a^4)^b}{a^b}=\left(\frac{81a^4}{a}\right)^b=(81a^3)^b$ , and since $x>0$ , $x=81a^3$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 9 hard

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $5$ and its exponent is multiplied by $3$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $5$ and the exponent multiplied by $3$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $5$ and its exponent multiplied by $3$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

5

gives

5a

; multiplying the exponent by

3

gives

3b

.

S=(5a)^{3b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

3b

as

3\times b

and use the power-of-a-power rule, then raise

5a

to the power

3

.

(5a)^{3b}=\{(5a)^3\}^b=(125a^3)^b

(5a)^3=5^3a^3=125a^3

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(125a^3)^b=(ax)^b \Rightarrow 125a^3=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{125a^3}{a}=125a^2

a^3\div a=a^{3-1}=a^2

.

Answer:

x=125a^2

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=10^3=1000$ , and $x=125\times2^2=500$ gives $a^b\times x^b=2\times500=1000$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(125a^3)^b}{a^b}=\left(\frac{125a^3}{a}\right)^b=(125a^2)^b$ , and since $x>0$ , $x=125a^2$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 10 hard

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $2$ and its exponent is multiplied by $5$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $2$ and the exponent multiplied by $5$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $2$ and its exponent multiplied by $5$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

2

gives

2a

; multiplying the exponent by

5

gives

5b

.

S=(2a)^{5b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

5b

as

5\times b

and use the power-of-a-power rule, then raise

2a

to the power

5

.

(2a)^{5b}=\{(2a)^5\}^b=(32a^5)^b

(2a)^5=2^5a^5=32a^5

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(32a^5)^b=(ax)^b \Rightarrow 32a^5=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{32a^5}{a}=32a^4

a^5\div a=a^{5-1}=a^4

.

Answer:

x=32a^4

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=4^5=1024$ , and $x=32\times2^4=512$ gives $a^b\times x^b=2\times512=1024$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(32a^5)^b}{a^b}=\left(\frac{32a^5}{a}\right)^b=(32a^4)^b$ , and since $x>0$ , $x=32a^4$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 11 hard

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $5$ and its exponent is multiplied by $2$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $5$ and the exponent multiplied by $2$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $5$ and its exponent multiplied by $2$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

5

gives

5a

; multiplying the exponent by

2

gives

2b

.

S=(5a)^{2b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

2b

as

2\times b

and use the power-of-a-power rule, then raise

5a

to the power

2

.

(5a)^{2b}=\{(5a)^2\}^b=(25a^2)^b

(5a)^2=5^2a^2=25a^2

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(25a^2)^b=(ax)^b \Rightarrow 25a^2=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{25a^2}{a}=25a

a^2\div a=a^{2-1}=a

.

Answer:

x=25a

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=10^2=100$ , and $x=25\times2=50$ gives $a^b\times x^b=2\times50=100$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(25a^2)^b}{a^b}=\left(\frac{25a^2}{a}\right)^b=(25a)^b$ , and since $x>0$ , $x=25a$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Variant 12 hard

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when its base is multiplied by $6$ and its exponent is multiplied by $2$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two descriptions give the same number $S$ . One says $S$ is $a^b$ with the base multiplied by $6$ and the exponent multiplied by $2$ ; the other says $S$ is $a^b$ times $x^b$ . Find $x$ in terms of $a$ .

Givens

$S$ is $a^b$ with its base multiplied by $6$ and its exponent multiplied by $2$ .
$S=a^b\times x^b$ .
$a>0$ , $b>0$ , $x>0$ .

Unknowns

$x$ written in terms of $a$ .

Constraints

All bases are positive and the exponent $b$ is positive, so two $b$ -th powers are equal only when their bases are equal.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #15 Organize Information in More Ways

Turn each sentence into an expression for $S$ , then use the exponent laws to write both as a power with the same exponent $b$ . Once the exponents match, the bases must match.

3 · Execute5 carry out the plan

1Write the first description

#13 Convert to Algebra 6.EE.A.2

Multiplying the base of

a^b

by

6

gives

6a

; multiplying the exponent by

2

gives

2b

.

S=(6a)^{2b}

This is just a translation of the words into symbols.

2Make the exponent $b$ appear

#15 Organize Information in More Ways 8.EE.A.1

Split the exponent

2b

as

2\times b

and use the power-of-a-power rule, then raise

6a

to the power

2

.

(6a)^{2b}=\{(6a)^2\}^b=(36a^2)^b

(6a)^2=6^2a^2=36a^2

.

3Write the second description

#15 Organize Information in More Ways 8.EE.A.1

Two powers with the same exponent multiply into one power of the product.

S=a^b\times x^b=(ax)^b

(ab)^n=a^nb^n

read backwards.

4Compare the bases

#13 Convert to Algebra 8.EE.A.1

Now

S

is a

b

-th power in two ways. The bases are positive and

b>0

, so a bigger base would give a bigger power; equal powers need equal bases.

(36a^2)^b=(ax)^b \Rightarrow 36a^2=ax

Same exponent, same result, so same base.

5Solve for $x$

#13 Convert to Algebra 8.EE.A.1

Divide both sides by

a

, which is not

0

.

x=\frac{36a^2}{a}=36a

a^2\div a=a^{2-1}=a

.

Answer:

x=36a

4 · Reviewdoes it hold up?

Try $a=2$ , $b=1$ : the first description gives $S=12^2=144$ , and $x=36\times2=72$ gives $a^b\times x^b=2\times72=144$ . They agree.

Another way: Divide instead of comparing: $x^b=\frac{S}{a^b}=\frac{(36a^2)^b}{a^b}=\left(\frac{36a^2}{a}\right)^b=(36a)^b$ , and since $x>0$ , $x=36a$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Power of a power, power of a product, and dividing powers of $a$.
6.EE.A.2 Write, read, and evaluate expressions with variables — Writing each sentence as an expression for $S$.

💡Takeaway. When two powers share an exponent, line them up and just compare the bases.

Same exponent, so match the bases · 12 practice problems

Generated variants — 12

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Givens

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Constraints

Givens

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Givens

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Where next?