Practice · Solve for the building blocks, then rebuild · Sensim Math

Variant 1 easy

If $a=3^{x+1}$ and $b=2^{x+1}$ , express $6^{2x}$ in terms of $a$ and $b$ . (You may use the fact that $6^2=36$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+1}$ and $b=2^{x+1}$ , write $6^{2x}$ using only $a$ and $b$ .

Givens

$a=3^{x+1}$
$b=2^{x+1}$
$6^2=36$

Unknowns

$6^{2x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$6$ is built from $2$ s and $3$ s, so $6^{2x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $6^{2x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+1

in the exponent.

a=3^{x+1}=3\times3^x \Rightarrow 3^x=\frac{a}{3}

3^{x+1}=3^x\times3

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+1

in the exponent.

b=2^{x+1}=2\times2^x \Rightarrow 2^x=\frac{b}{2}

Same idea with base

2

.

3Break $6^{2x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

6=2\times3

, then group the exponents around

x

.

6^{2x}=(2\times3)^{2x}=2^{2x}\times3^{2x}=(2^x)^2\times(3^x)^2

2^{2x}=(2^x)^2

and

3^{2x}=(3^x)^2

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{2}\right)^2\times\left(\frac{a}{3}\right)^2=\frac{b^2}{4}\times\frac{a^2}{9}

2^2=4

and

3^2=9

.

5Combine the denominators

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

4\times9=2^2\times3^2=6^2

, which the problem says is

36

.

\frac{a^2b^2}{2^2\times3^2}=\frac{a^2b^2}{6^2}=\frac{a^2b^2}{36}

2^2\times3^2=(2\times3)^2

.

Answer:

\frac{a^2b^2}{36}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=9$ , $b=4$ , and $6^2=36$ . The formula gives $\frac{9^2\times4^2}{36}=\frac{81\times16}{36}=\frac{1296}{36}=36$ . It matches.

Another way: Write $a^2b^2$ directly: $a^2b^2=3^{2x+2}\times2^{2x+2}=(2^{2x}\times3^{2x})\times(2^2\times3^2)=6^{2x}\times36$ , so $6^{2x}=\frac{a^2b^2}{36}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 2 easy

If $a=3^{x+1}$ and $b=2^{x+1}$ , express $6^{3x}$ in terms of $a$ and $b$ . (You may use the fact that $6^3=216$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+1}$ and $b=2^{x+1}$ , write $6^{3x}$ using only $a$ and $b$ .

Givens

$a=3^{x+1}$
$b=2^{x+1}$
$6^3=216$

Unknowns

$6^{3x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$6$ is built from $2$ s and $3$ s, so $6^{3x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $6^{3x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+1

in the exponent.

a=3^{x+1}=3\times3^x \Rightarrow 3^x=\frac{a}{3}

3^{x+1}=3^x\times3

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+1

in the exponent.

b=2^{x+1}=2\times2^x \Rightarrow 2^x=\frac{b}{2}

Same idea with base

2

.

3Break $6^{3x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

6=2\times3

, then group the exponents around

x

.

6^{3x}=(2\times3)^{3x}=2^{3x}\times3^{3x}=(2^x)^3\times(3^x)^3

2^{3x}=(2^x)^3

and

3^{3x}=(3^x)^3

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{2}\right)^3\times\left(\frac{a}{3}\right)^3=\frac{b^3}{8}\times\frac{a^3}{27}

2^3=8

and

3^3=27

.

5Combine the denominators

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

8\times27=2^3\times3^3=6^3

, which the problem says is

216

.

\frac{a^3b^3}{2^3\times3^3}=\frac{a^3b^3}{6^3}=\frac{a^3b^3}{216}

2^3\times3^3=(2\times3)^3

.

Answer:

\frac{a^3b^3}{216}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=9$ , $b=4$ , and $6^3=216$ . The formula gives $\frac{9^3\times4^3}{216}=\frac{729\times64}{216}=\frac{46656}{216}=216$ . It matches.

Another way: Write $a^3b^3$ directly: $a^3b^3=3^{3x+3}\times2^{3x+3}=(2^{3x}\times3^{3x})\times(2^3\times3^3)=6^{3x}\times216$ , so $6^{3x}=\frac{a^3b^3}{216}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 3 easy

If $a=3^{x+2}$ and $b=2^{x+2}$ , express $36^{x}$ in terms of $a$ and $b$ . (You may use the fact that $6^4=1296$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+2}$ and $b=2^{x+2}$ , write $36^{x}$ using only $a$ and $b$ .

Givens

$a=3^{x+2}$
$b=2^{x+2}$
$6^4=1296$

Unknowns

$36^{x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$36$ is built from $2$ s and $3$ s, so $36^{x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $36^{x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+2

in the exponent.

a=3^{x+2}=3^x\times3^2=9\times3^x \Rightarrow 3^x=\frac{a}{9}

3^{x+2}=3^x\times3^2

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+2

in the exponent.

b=2^{x+2}=2^x\times2^2=4\times2^x \Rightarrow 2^x=\frac{b}{4}

Same idea with base

2

.

3Break $36^{x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

36=2^2\times3^2

, then group the exponents around

x

.

36^{x}=(2^2\times3^2)^{x}=2^{2x}\times3^{2x}=(2^x)^2\times(3^x)^2

2^{2x}=(2^x)^2

and

3^{2x}=(3^x)^2

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{4}\right)^2\times\left(\frac{a}{9}\right)^2=\frac{b^2}{16}\times\frac{a^2}{81}

4^2=16

and

9^2=81

.

5Combine the denominators

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

16\times81=2^4\times3^4=6^4

, which the problem says is

1296

.

\frac{a^2b^2}{2^4\times3^4}=\frac{a^2b^2}{6^4}=\frac{a^2b^2}{1296}

2^4\times3^4=(2\times3)^4

.

Answer:

\frac{a^2b^2}{1296}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=27$ , $b=8$ , and $36^1=36$ . The formula gives $\frac{27^2\times8^2}{1296}=\frac{729\times64}{1296}=\frac{46656}{1296}=36$ . It matches.

Another way: Write $a^2b^2$ directly: $a^2b^2=3^{2x+4}\times2^{2x+4}=(2^{2x}\times3^{2x})\times(2^4\times3^4)=36^{x}\times1296$ , so $36^{x}=\frac{a^2b^2}{1296}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 4 medium

If $a=3^{x+1}$ and $b=2^{x+1}$ , express $36^{2x}$ in terms of $a$ and $b$ . (You may use the fact that $6^4=1296$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+1}$ and $b=2^{x+1}$ , write $36^{2x}$ using only $a$ and $b$ .

Givens

$a=3^{x+1}$
$b=2^{x+1}$
$6^4=1296$

Unknowns

$36^{2x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$36$ is built from $2$ s and $3$ s, so $36^{2x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $36^{2x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+1

in the exponent.

a=3^{x+1}=3\times3^x \Rightarrow 3^x=\frac{a}{3}

3^{x+1}=3^x\times3

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+1

in the exponent.

b=2^{x+1}=2\times2^x \Rightarrow 2^x=\frac{b}{2}

Same idea with base

2

.

3Break $36^{2x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

36=2^2\times3^2

, then group the exponents around

x

.

36^{2x}=(2^2\times3^2)^{2x}=2^{4x}\times3^{4x}=(2^x)^4\times(3^x)^4

2^{4x}=(2^x)^4

and

3^{4x}=(3^x)^4

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{2}\right)^4\times\left(\frac{a}{3}\right)^4=\frac{b^4}{16}\times\frac{a^4}{81}

2^4=16

and

3^4=81

.

5Combine the denominators

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

16\times81=2^4\times3^4=6^4

, which the problem says is

1296

.

\frac{a^4b^4}{2^4\times3^4}=\frac{a^4b^4}{6^4}=\frac{a^4b^4}{1296}

2^4\times3^4=(2\times3)^4

.

Answer:

\frac{a^4b^4}{1296}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=9$ , $b=4$ , and $36^2=1296$ . The formula gives $\frac{9^4\times4^4}{1296}=\frac{6561\times256}{1296}=\frac{1679616}{1296}=1296$ . It matches.

Another way: Write $a^4b^4$ directly: $a^4b^4=3^{4x+4}\times2^{4x+4}=(2^{4x}\times3^{4x})\times(2^4\times3^4)=36^{2x}\times1296$ , so $36^{2x}=\frac{a^4b^4}{1296}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 5 medium

If $a=3^{x+1}$ and $b=2^{x+1}$ , express $6^{4x}$ in terms of $a$ and $b$ . (You may use the fact that $6^4=1296$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+1}$ and $b=2^{x+1}$ , write $6^{4x}$ using only $a$ and $b$ .

Givens

$a=3^{x+1}$
$b=2^{x+1}$
$6^4=1296$

Unknowns

$6^{4x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$6$ is built from $2$ s and $3$ s, so $6^{4x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $6^{4x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+1

in the exponent.

a=3^{x+1}=3\times3^x \Rightarrow 3^x=\frac{a}{3}

3^{x+1}=3^x\times3

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+1

in the exponent.

b=2^{x+1}=2\times2^x \Rightarrow 2^x=\frac{b}{2}

Same idea with base

2

.

3Break $6^{4x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

6=2\times3

, then group the exponents around

x

.

6^{4x}=(2\times3)^{4x}=2^{4x}\times3^{4x}=(2^x)^4\times(3^x)^4

2^{4x}=(2^x)^4

and

3^{4x}=(3^x)^4

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{2}\right)^4\times\left(\frac{a}{3}\right)^4=\frac{b^4}{16}\times\frac{a^4}{81}

2^4=16

and

3^4=81

.

5Combine the denominators

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

16\times81=2^4\times3^4=6^4

, which the problem says is

1296

.

\frac{a^4b^4}{2^4\times3^4}=\frac{a^4b^4}{6^4}=\frac{a^4b^4}{1296}

2^4\times3^4=(2\times3)^4

.

Answer:

\frac{a^4b^4}{1296}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=9$ , $b=4$ , and $6^4=1296$ . The formula gives $\frac{9^4\times4^4}{1296}=\frac{6561\times256}{1296}=\frac{1679616}{1296}=1296$ . It matches.

Another way: Write $a^4b^4$ directly: $a^4b^4=3^{4x+4}\times2^{4x+4}=(2^{4x}\times3^{4x})\times(2^4\times3^4)=6^{4x}\times1296$ , so $6^{4x}=\frac{a^4b^4}{1296}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 6 easy

If $a=3^{x+2}$ and $b=2^{x+1}$ , express $12^{2x}$ in terms of $a$ and $b$ . (You may use the fact that $6^4=1296$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+2}$ and $b=2^{x+1}$ , write $12^{2x}$ using only $a$ and $b$ .

Givens

$a=3^{x+2}$
$b=2^{x+1}$
$6^4=1296$

Unknowns

$12^{2x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$12$ is built from $2$ s and $3$ s, so $12^{2x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $12^{2x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+2

in the exponent.

a=3^{x+2}=3^x\times3^2=9\times3^x \Rightarrow 3^x=\frac{a}{9}

3^{x+2}=3^x\times3^2

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+1

in the exponent.

b=2^{x+1}=2\times2^x \Rightarrow 2^x=\frac{b}{2}

Same idea with base

2

.

3Break $12^{2x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

12=2^2\times3

, then group the exponents around

x

.

12^{2x}=(2^2\times3)^{2x}=2^{4x}\times3^{2x}=(2^x)^4\times(3^x)^2

2^{4x}=(2^x)^4

and

3^{2x}=(3^x)^2

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{2}\right)^4\times\left(\frac{a}{9}\right)^2=\frac{b^4}{16}\times\frac{a^2}{81}

2^4=16

and

9^2=81

.

5Combine the denominators

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

16\times81=2^4\times3^4=6^4

, which the problem says is

1296

.

\frac{a^2b^4}{2^4\times3^4}=\frac{a^2b^4}{6^4}=\frac{a^2b^4}{1296}

2^4\times3^4=(2\times3)^4

.

Answer:

\frac{a^2b^4}{1296}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=27$ , $b=4$ , and $12^2=144$ . The formula gives $\frac{27^2\times4^4}{1296}=\frac{729\times256}{1296}=\frac{186624}{1296}=144$ . It matches.

Another way: Write $a^2b^4$ directly: $a^2b^4=3^{2x+4}\times2^{4x+4}=(2^{4x}\times3^{2x})\times(2^4\times3^4)=12^{2x}\times1296$ , so $12^{2x}=\frac{a^2b^4}{1296}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 7 medium

If $a=3^{x+2}$ and $b=2^{x+2}$ , express $6^{2x}$ in terms of $a$ and $b$ . (You may use the fact that $6^4=1296$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+2}$ and $b=2^{x+2}$ , write $6^{2x}$ using only $a$ and $b$ .

Givens

$a=3^{x+2}$
$b=2^{x+2}$
$6^4=1296$

Unknowns

$6^{2x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$6$ is built from $2$ s and $3$ s, so $6^{2x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $6^{2x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+2

in the exponent.

a=3^{x+2}=3^x\times3^2=9\times3^x \Rightarrow 3^x=\frac{a}{9}

3^{x+2}=3^x\times3^2

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+2

in the exponent.

b=2^{x+2}=2^x\times2^2=4\times2^x \Rightarrow 2^x=\frac{b}{4}

Same idea with base

2

.

3Break $6^{2x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

6=2\times3

, then group the exponents around

x

.

6^{2x}=(2\times3)^{2x}=2^{2x}\times3^{2x}=(2^x)^2\times(3^x)^2

2^{2x}=(2^x)^2

and

3^{2x}=(3^x)^2

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{4}\right)^2\times\left(\frac{a}{9}\right)^2=\frac{b^2}{16}\times\frac{a^2}{81}

4^2=16

and

9^2=81

.

5Combine the denominators

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

16\times81=2^4\times3^4=6^4

, which the problem says is

1296

.

\frac{a^2b^2}{2^4\times3^4}=\frac{a^2b^2}{6^4}=\frac{a^2b^2}{1296}

2^4\times3^4=(2\times3)^4

.

Answer:

\frac{a^2b^2}{1296}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=27$ , $b=8$ , and $6^2=36$ . The formula gives $\frac{27^2\times8^2}{1296}=\frac{729\times64}{1296}=\frac{46656}{1296}=36$ . It matches.

Another way: Write $a^2b^2$ directly: $a^2b^2=3^{2x+4}\times2^{2x+4}=(2^{2x}\times3^{2x})\times(2^4\times3^4)=6^{2x}\times1296$ , so $6^{2x}=\frac{a^2b^2}{1296}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 8 medium

If $a=3^{x+1}$ and $b=2^{x+2}$ , express $18^{2x}$ in terms of $a$ and $b$ . (You may use the fact that $6^4=1296$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+1}$ and $b=2^{x+2}$ , write $18^{2x}$ using only $a$ and $b$ .

Givens

$a=3^{x+1}$
$b=2^{x+2}$
$6^4=1296$

Unknowns

$18^{2x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$18$ is built from $2$ s and $3$ s, so $18^{2x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $18^{2x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+1

in the exponent.

a=3^{x+1}=3\times3^x \Rightarrow 3^x=\frac{a}{3}

3^{x+1}=3^x\times3

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+2

in the exponent.

b=2^{x+2}=2^x\times2^2=4\times2^x \Rightarrow 2^x=\frac{b}{4}

Same idea with base

2

.

3Break $18^{2x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

18=2\times3^2

, then group the exponents around

x

.

18^{2x}=(2\times3^2)^{2x}=2^{2x}\times3^{4x}=(2^x)^2\times(3^x)^4

2^{2x}=(2^x)^2

and

3^{4x}=(3^x)^4

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{4}\right)^2\times\left(\frac{a}{3}\right)^4=\frac{b^2}{16}\times\frac{a^4}{81}

4^2=16

and

3^4=81

.

5Combine the denominators

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

16\times81=2^4\times3^4=6^4

, which the problem says is

1296

.

\frac{a^4b^2}{2^4\times3^4}=\frac{a^4b^2}{6^4}=\frac{a^4b^2}{1296}

2^4\times3^4=(2\times3)^4

.

Answer:

\frac{a^4b^2}{1296}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=9$ , $b=8$ , and $18^2=324$ . The formula gives $\frac{9^4\times8^2}{1296}=\frac{6561\times64}{1296}=\frac{419904}{1296}=324$ . It matches.

Another way: Write $a^4b^2$ directly: $a^4b^2=3^{4x+4}\times2^{2x+4}=(2^{2x}\times3^{4x})\times(2^4\times3^4)=18^{2x}\times1296$ , so $18^{2x}=\frac{a^4b^2}{1296}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 9 hard

If $a=3^{x+1}$ and $b=2^{x+3}$ , express $54^{2x}$ in terms of $a$ and $b$ . (You may use the fact that $6^6=46656$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+1}$ and $b=2^{x+3}$ , write $54^{2x}$ using only $a$ and $b$ .

Givens

$a=3^{x+1}$
$b=2^{x+3}$
$6^6=46656$

Unknowns

$54^{2x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$54$ is built from $2$ s and $3$ s, so $54^{2x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $54^{2x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+1

in the exponent.

a=3^{x+1}=3\times3^x \Rightarrow 3^x=\frac{a}{3}

3^{x+1}=3^x\times3

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+3

in the exponent.

b=2^{x+3}=2^x\times2^3=8\times2^x \Rightarrow 2^x=\frac{b}{8}

Same idea with base

2

.

3Break $54^{2x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

54=2\times3^3

, then group the exponents around

x

.

54^{2x}=(2\times3^3)^{2x}=2^{2x}\times3^{6x}=(2^x)^2\times(3^x)^6

2^{2x}=(2^x)^2

and

3^{6x}=(3^x)^6

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{8}\right)^2\times\left(\frac{a}{3}\right)^6=\frac{b^2}{64}\times\frac{a^6}{729}

8^2=64

and

3^6=729

.

5Combine the denominators

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

64\times729=2^6\times3^6=6^6

, which the problem says is

46656

.

\frac{a^6b^2}{2^6\times3^6}=\frac{a^6b^2}{6^6}=\frac{a^6b^2}{46656}

2^6\times3^6=(2\times3)^6

.

Answer:

\frac{a^6b^2}{46656}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=9$ , $b=16$ , and $54^2=2916$ . The formula gives $\frac{9^6\times16^2}{46656}=\frac{531441\times256}{46656}=\frac{136048896}{46656}=2916$ . It matches.

Another way: Write $a^6b^2$ directly: $a^6b^2=3^{6x+6}\times2^{2x+6}=(2^{2x}\times3^{6x})\times(2^6\times3^6)=54^{2x}\times46656$ , so $54^{2x}=\frac{a^6b^2}{46656}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 10 hard

If $a=3^{x+2}$ and $b=2^{x+1}$ , express $12^{3x}$ in terms of $a$ and $b$ . (You may use the fact that $6^6=46656$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+2}$ and $b=2^{x+1}$ , write $12^{3x}$ using only $a$ and $b$ .

Givens

$a=3^{x+2}$
$b=2^{x+1}$
$6^6=46656$

Unknowns

$12^{3x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$12$ is built from $2$ s and $3$ s, so $12^{3x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $12^{3x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+2

in the exponent.

a=3^{x+2}=3^x\times3^2=9\times3^x \Rightarrow 3^x=\frac{a}{9}

3^{x+2}=3^x\times3^2

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+1

in the exponent.

b=2^{x+1}=2\times2^x \Rightarrow 2^x=\frac{b}{2}

Same idea with base

2

.

3Break $12^{3x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

12=2^2\times3

, then group the exponents around

x

.

12^{3x}=(2^2\times3)^{3x}=2^{6x}\times3^{3x}=(2^x)^6\times(3^x)^3

2^{6x}=(2^x)^6

and

3^{3x}=(3^x)^3

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{2}\right)^6\times\left(\frac{a}{9}\right)^3=\frac{b^6}{64}\times\frac{a^3}{729}

2^6=64

and

9^3=729

.

5Combine the denominators

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

64\times729=2^6\times3^6=6^6

, which the problem says is

46656

.

\frac{a^3b^6}{2^6\times3^6}=\frac{a^3b^6}{6^6}=\frac{a^3b^6}{46656}

2^6\times3^6=(2\times3)^6

.

Answer:

\frac{a^3b^6}{46656}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=27$ , $b=4$ , and $12^3=1728$ . The formula gives $\frac{27^3\times4^6}{46656}=\frac{19683\times4096}{46656}=\frac{80621568}{46656}=1728$ . It matches.

Another way: Write $a^3b^6$ directly: $a^3b^6=3^{3x+6}\times2^{6x+6}=(2^{6x}\times3^{3x})\times(2^6\times3^6)=12^{3x}\times46656$ , so $12^{3x}=\frac{a^3b^6}{46656}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 11 hard

If $a=3^{x+1}$ and $b=2^{x+2}$ , express $18^{3x}$ in terms of $a$ and $b$ . (You may use the fact that $6^6=46656$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+1}$ and $b=2^{x+2}$ , write $18^{3x}$ using only $a$ and $b$ .

Givens

$a=3^{x+1}$
$b=2^{x+2}$
$6^6=46656$

Unknowns

$18^{3x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$18$ is built from $2$ s and $3$ s, so $18^{3x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $18^{3x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+1

in the exponent.

a=3^{x+1}=3\times3^x \Rightarrow 3^x=\frac{a}{3}

3^{x+1}=3^x\times3

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+2

in the exponent.

b=2^{x+2}=2^x\times2^2=4\times2^x \Rightarrow 2^x=\frac{b}{4}

Same idea with base

2

.

3Break $18^{3x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

18=2\times3^2

, then group the exponents around

x

.

18^{3x}=(2\times3^2)^{3x}=2^{3x}\times3^{6x}=(2^x)^3\times(3^x)^6

2^{3x}=(2^x)^3

and

3^{6x}=(3^x)^6

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{4}\right)^3\times\left(\frac{a}{3}\right)^6=\frac{b^3}{64}\times\frac{a^6}{729}

4^3=64

and

3^6=729

.

5Combine the denominators

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

64\times729=2^6\times3^6=6^6

, which the problem says is

46656

.

\frac{a^6b^3}{2^6\times3^6}=\frac{a^6b^3}{6^6}=\frac{a^6b^3}{46656}

2^6\times3^6=(2\times3)^6

.

Answer:

\frac{a^6b^3}{46656}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=9$ , $b=8$ , and $18^3=5832$ . The formula gives $\frac{9^6\times8^3}{46656}=\frac{531441\times512}{46656}=\frac{272097792}{46656}=5832$ . It matches.

Another way: Write $a^6b^3$ directly: $a^6b^3=3^{6x+6}\times2^{3x+6}=(2^{3x}\times3^{6x})\times(2^6\times3^6)=18^{3x}\times46656$ , so $18^{3x}=\frac{a^6b^3}{46656}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Variant 12 hard

If $a=3^{x+3}$ and $b=2^{x+1}$ , express $24^{2x}$ in terms of $a$ and $b$ . (You may use the fact that $6^6=46656$ .)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Given $a=3^{x+3}$ and $b=2^{x+1}$ , write $24^{2x}$ using only $a$ and $b$ .

Givens

$a=3^{x+3}$
$b=2^{x+1}$
$6^6=46656$

Unknowns

$24^{2x}$ in terms of $a$ and $b$ .

Constraints

The answer must not contain $x$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

$24$ is built from $2$ s and $3$ s, so $24^{2x}$ is built from $2^x$ and $3^x$ . First express those two building blocks with $a$ and $b$ , then assemble $24^{2x}$ from them.

3 · Execute5 carry out the plan

1Express $3^x$ with $a$

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

Split off the

+3

in the exponent.

a=3^{x+3}=3^x\times3^3=27\times3^x \Rightarrow 3^x=\frac{a}{27}

3^{x+3}=3^x\times3^3

by the product rule.

2Express $2^x$ with $b$

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

Split off the

+1

in the exponent.

b=2^{x+1}=2\times2^x \Rightarrow 2^x=\frac{b}{2}

Same idea with base

2

.

3Break $24^{2x}$ into the building blocks

#7 Identify Subproblems 8.EE.A.1

Use

24=2^3\times3

, then group the exponents around

x

.

24^{2x}=(2^3\times3)^{2x}=2^{6x}\times3^{2x}=(2^x)^6\times(3^x)^2

2^{6x}=(2^x)^6

and

3^{2x}=(3^x)^2

.

4Substitute

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

Replace

2^x

and

3^x

with the expressions from the first two steps.

\left(\frac{b}{2}\right)^6\times\left(\frac{a}{27}\right)^2=\frac{b^6}{64}\times\frac{a^2}{729}

2^6=64

and

27^2=729

.

5Combine the denominators

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

64\times729=2^6\times3^6=6^6

, which the problem says is

46656

.

\frac{a^2b^6}{2^6\times3^6}=\frac{a^2b^6}{6^6}=\frac{a^2b^6}{46656}

2^6\times3^6=(2\times3)^6

.

Answer:

\frac{a^2b^6}{46656}

4 · Reviewdoes it hold up?

Test $x=1$ : $a=81$ , $b=4$ , and $24^2=576$ . The formula gives $\frac{81^2\times4^6}{46656}=\frac{6561\times4096}{46656}=\frac{26873856}{46656}=576$ . It matches.

Another way: Write $a^2b^6$ directly: $a^2b^6=3^{2x+6}\times2^{6x+6}=(2^{6x}\times3^{2x})\times(2^6\times3^6)=24^{2x}\times46656$ , so $24^{2x}=\frac{a^2b^6}{46656}$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting and regrouping exponents with the product and power rules.
6.EE.A.2 Write, read, and evaluate expressions with variables — Substituting expressions in $a$ and $b$ for $2^x$ and $3^x$.

💡Takeaway. Find the small pieces (

2^x

,

3^x

) in the given letters first; then any power made of them follows.

Solve for the building blocks, then rebuild · 12 practice problems

Generated variants — 12

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