Practice · Match the exponents, then compare bases · Sensim Math

Variant 1 easy

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{14}$ , $3^{10}$ , $5^6$ , $7^4$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{14}$ , $3^{10}$ , $5^6$ , and $7^4$ , find the largest and the smallest.

Givens

The numbers $2^{14}$ , $3^{10}$ , $5^6$ , $7^4$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $14$ , $10$ , $6$ , and $4$ are all multiples of $2$ . Rewrite every number as something to the power $2$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

2

, so aim for the form

(\ )^{2}

.

14=7\times2,\ 10=5\times2,\ 6=3\times2,\ 4=2\times2

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{14}=(2^7)^{2}=128^2,\ 3^{10}=(3^5)^{2}=243^2,\ 5^6=(5^3)^{2}=125^2,\ 7^4=(7^2)^{2}=49^2

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

49<125<128<243 \Rightarrow 49^2<125^2<128^2<243^2

Multiplying 2 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

2

with the original number.

7^4<5^6<2^{14}<3^{10}

128^2

is

2^{14}

, and so on.

Answer:

3^{10}

,

7^4

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{14}\approx1.6\times10^4$ , $3^{10}\approx5.9\times10^4$ , $5^6\approx1.6\times10^4$ , $7^4\approx2.4\times10^3$ . The biggest is $3^{10}$ and the smallest is $7^4$ .

Another way: Compare pairs by dividing: for example $\frac{3^{10}}{2^{14}}=\frac{243^2}{128^2}=\left(\frac{243}{128}\right)^{2}>1$ , so $3^{10}>2^{14}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $2$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 2 easy

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{21}$ , $3^{12}$ , $5^9$ , $11^6$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{21}$ , $3^{12}$ , $5^9$ , and $11^6$ , find the largest and the smallest.

Givens

The numbers $2^{21}$ , $3^{12}$ , $5^9$ , $11^6$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $21$ , $12$ , $9$ , and $6$ are all multiples of $3$ . Rewrite every number as something to the power $3$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

3

, so aim for the form

(\ )^{3}

.

21=7\times3,\ 12=4\times3,\ 9=3\times3,\ 6=2\times3

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{21}=(2^7)^{3}=128^3,\ 3^{12}=(3^4)^{3}=81^3,\ 5^9=(5^3)^{3}=125^3,\ 11^6=(11^2)^{3}=121^3

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

81<121<125<128 \Rightarrow 81^3<121^3<125^3<128^3

Multiplying 3 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

3

with the original number.

3^{12}<11^6<5^9<2^{21}

128^3

is

2^{21}

, and so on.

Answer:

2^{21}

,

3^{12}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{21}\approx2.1\times10^6$ , $3^{12}\approx5.3\times10^5$ , $5^9\approx2.0\times10^6$ , $11^6\approx1.8\times10^6$ . The biggest is $2^{21}$ and the smallest is $3^{12}$ .

Another way: Compare pairs by dividing: for example $\frac{2^{21}}{5^9}=\frac{128^3}{125^3}=\left(\frac{128}{125}\right)^{3}>1$ , so $2^{21}>5^9$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $3$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 3 easy

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{24}$ , $3^{18}$ , $6^{12}$ , $20^6$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{24}$ , $3^{18}$ , $6^{12}$ , and $20^6$ , find the largest and the smallest.

Givens

The numbers $2^{24}$ , $3^{18}$ , $6^{12}$ , $20^6$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $24$ , $18$ , $12$ , and $6$ are all multiples of $6$ . Rewrite every number as something to the power $6$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

6

, so aim for the form

(\ )^{6}

.

24=4\times6,\ 18=3\times6,\ 12=2\times6

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{24}=(2^4)^{6}=16^6,\ 3^{18}=(3^3)^{6}=27^6,\ 6^{12}=(6^2)^{6}=36^6

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

16<20<27<36 \Rightarrow 16^6<20^6<27^6<36^6

Multiplying 6 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

6

with the original number.

2^{24}<20^6<3^{18}<6^{12}

16^6

is

2^{24}

, and so on.

Answer:

6^{12}

,

2^{24}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{24}\approx1.7\times10^7$ , $3^{18}\approx3.9\times10^8$ , $6^{12}\approx2.2\times10^9$ , $20^6\approx6.4\times10^7$ . The biggest is $6^{12}$ and the smallest is $2^{24}$ .

Another way: Compare pairs by dividing: for example $\frac{6^{12}}{3^{18}}=\frac{36^6}{27^6}=\left(\frac{36}{27}\right)^{6}>1$ , so $6^{12}>3^{18}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $6$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 4 easy

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{20}$ , $3^{15}$ , $5^{10}$ , $26^5$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{20}$ , $3^{15}$ , $5^{10}$ , and $26^5$ , find the largest and the smallest.

Givens

The numbers $2^{20}$ , $3^{15}$ , $5^{10}$ , $26^5$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $20$ , $15$ , $10$ , and $5$ are all multiples of $5$ . Rewrite every number as something to the power $5$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

5

, so aim for the form

(\ )^{5}

.

20=4\times5,\ 15=3\times5,\ 10=2\times5

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{20}=(2^4)^{5}=16^5,\ 3^{15}=(3^3)^{5}=27^5,\ 5^{10}=(5^2)^{5}=25^5

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

16<25<26<27 \Rightarrow 16^5<25^5<26^5<27^5

Multiplying 5 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

5

with the original number.

2^{20}<5^{10}<26^5<3^{15}

16^5

is

2^{20}

, and so on.

Answer:

3^{15}

,

2^{20}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{20}\approx1.0\times10^6$ , $3^{15}\approx1.4\times10^7$ , $5^{10}\approx9.8\times10^6$ , $26^5\approx1.2\times10^7$ . The biggest is $3^{15}$ and the smallest is $2^{20}$ .

Another way: Compare pairs by dividing: for example $\frac{3^{15}}{26^5}=\frac{27^5}{26^5}=\left(\frac{27}{26}\right)^{5}>1$ , so $3^{15}>26^5$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $5$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 5 medium

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{20}$ , $3^{12}$ , $6^8$ , $30^4$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{20}$ , $3^{12}$ , $6^8$ , and $30^4$ , find the largest and the smallest.

Givens

The numbers $2^{20}$ , $3^{12}$ , $6^8$ , $30^4$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $20$ , $12$ , $8$ , and $4$ are all multiples of $4$ . Rewrite every number as something to the power $4$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

4

, so aim for the form

(\ )^{4}

.

20=5\times4,\ 12=3\times4,\ 8=2\times4

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{20}=(2^5)^{4}=32^4,\ 3^{12}=(3^3)^{4}=27^4,\ 6^8=(6^2)^{4}=36^4

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

27<30<32<36 \Rightarrow 27^4<30^4<32^4<36^4

Multiplying 4 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

4

with the original number.

3^{12}<30^4<2^{20}<6^8

32^4

is

2^{20}

, and so on.

Answer:

6^8

,

3^{12}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{20}\approx1.0\times10^6$ , $3^{12}\approx5.3\times10^5$ , $6^8\approx1.7\times10^6$ , $30^4\approx8.1\times10^5$ . The biggest is $6^8$ and the smallest is $3^{12}$ .

Another way: Compare pairs by dividing: for example $\frac{6^8}{2^{20}}=\frac{36^4}{32^4}=\left(\frac{36}{32}\right)^{4}>1$ , so $6^8>2^{20}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $4$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 6 medium

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{40}$ , $3^{30}$ , $5^{20}$ , $15^{10}$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{40}$ , $3^{30}$ , $5^{20}$ , and $15^{10}$ , find the largest and the smallest.

Givens

The numbers $2^{40}$ , $3^{30}$ , $5^{20}$ , $15^{10}$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $40$ , $30$ , $20$ , and $10$ are all multiples of $10$ . Rewrite every number as something to the power $10$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

10

, so aim for the form

(\ )^{10}

.

40=4\times10,\ 30=3\times10,\ 20=2\times10

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{40}=(2^4)^{10}=16^{10},\ 3^{30}=(3^3)^{10}=27^{10},\ 5^{20}=(5^2)^{10}=25^{10}

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

15<16<25<27 \Rightarrow 15^{10}<16^{10}<25^{10}<27^{10}

Multiplying 10 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

10

with the original number.

15^{10}<2^{40}<5^{20}<3^{30}

16^{10}

is

2^{40}

, and so on.

Answer:

3^{30}

,

15^{10}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{40}\approx1.1\times10^{12}$ , $3^{30}\approx2.1\times10^{14}$ , $5^{20}\approx9.5\times10^{13}$ , $15^{10}\approx5.8\times10^{11}$ . The biggest is $3^{30}$ and the smallest is $15^{10}$ .

Another way: Compare pairs by dividing: for example $\frac{3^{30}}{5^{20}}=\frac{27^{10}}{25^{10}}=\left(\frac{27}{25}\right)^{10}>1$ , so $3^{30}>5^{20}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $10$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 7 medium

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{40}$ , $3^{30}$ , $6^{20}$ , $30^{10}$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{40}$ , $3^{30}$ , $6^{20}$ , and $30^{10}$ , find the largest and the smallest.

Givens

The numbers $2^{40}$ , $3^{30}$ , $6^{20}$ , $30^{10}$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $40$ , $30$ , $20$ , and $10$ are all multiples of $10$ . Rewrite every number as something to the power $10$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

10

, so aim for the form

(\ )^{10}

.

40=4\times10,\ 30=3\times10,\ 20=2\times10

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{40}=(2^4)^{10}=16^{10},\ 3^{30}=(3^3)^{10}=27^{10},\ 6^{20}=(6^2)^{10}=36^{10}

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

16<27<30<36 \Rightarrow 16^{10}<27^{10}<30^{10}<36^{10}

Multiplying 10 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

10

with the original number.

2^{40}<3^{30}<30^{10}<6^{20}

16^{10}

is

2^{40}

, and so on.

Answer:

6^{20}

,

2^{40}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{40}\approx1.1\times10^{12}$ , $3^{30}\approx2.1\times10^{14}$ , $6^{20}\approx3.7\times10^{15}$ , $30^{10}\approx5.9\times10^{14}$ . The biggest is $6^{20}$ and the smallest is $2^{40}$ .

Another way: Compare pairs by dividing: for example $\frac{6^{20}}{30^{10}}=\frac{36^{10}}{30^{10}}=\left(\frac{36}{30}\right)^{10}>1$ , so $6^{20}>30^{10}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $10$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 8 medium

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{45}$ , $3^{30}$ , $5^{15}$ , $6^{15}$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{45}$ , $3^{30}$ , $5^{15}$ , and $6^{15}$ , find the largest and the smallest.

Givens

The numbers $2^{45}$ , $3^{30}$ , $5^{15}$ , $6^{15}$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases are all different and the exponents are not all the same, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $45$ , $30$ , $15$ , and $15$ are all multiples of $15$ . Rewrite every number as something to the power $15$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

15

, so aim for the form

(\ )^{15}

.

45=3\times15,\ 30=2\times15

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{45}=(2^3)^{15}=8^{15},\ 3^{30}=(3^2)^{15}=9^{15}

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

5<6<8<9 \Rightarrow 5^{15}<6^{15}<8^{15}<9^{15}

Multiplying 15 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

15

with the original number.

5^{15}<6^{15}<2^{45}<3^{30}

8^{15}

is

2^{45}

, and so on.

Answer:

3^{30}

,

5^{15}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{45}\approx3.5\times10^{13}$ , $3^{30}\approx2.1\times10^{14}$ , $5^{15}\approx3.1\times10^{10}$ , $6^{15}\approx4.7\times10^{11}$ . The biggest is $3^{30}$ and the smallest is $5^{15}$ .

Another way: Compare pairs by dividing: for example $\frac{3^{30}}{2^{45}}=\frac{9^{15}}{8^{15}}=\left(\frac{9}{8}\right)^{15}>1$ , so $3^{30}>2^{45}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $15$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 9 hard

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{48}$ , $3^{36}$ , $5^{24}$ , $24^{12}$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{48}$ , $3^{36}$ , $5^{24}$ , and $24^{12}$ , find the largest and the smallest.

Givens

The numbers $2^{48}$ , $3^{36}$ , $5^{24}$ , $24^{12}$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $48$ , $36$ , $24$ , and $12$ are all multiples of $12$ . Rewrite every number as something to the power $12$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

12

, so aim for the form

(\ )^{12}

.

48=4\times12,\ 36=3\times12,\ 24=2\times12

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{48}=(2^4)^{12}=16^{12},\ 3^{36}=(3^3)^{12}=27^{12},\ 5^{24}=(5^2)^{12}=25^{12}

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

16<24<25<27 \Rightarrow 16^{12}<24^{12}<25^{12}<27^{12}

Multiplying 12 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

12

with the original number.

2^{48}<24^{12}<5^{24}<3^{36}

16^{12}

is

2^{48}

, and so on.

Answer:

3^{36}

,

2^{48}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{48}\approx2.8\times10^{14}$ , $3^{36}\approx1.5\times10^{17}$ , $5^{24}\approx6.0\times10^{16}$ , $24^{12}\approx3.7\times10^{16}$ . The biggest is $3^{36}$ and the smallest is $2^{48}$ .

Another way: Compare pairs by dividing: for example $\frac{3^{36}}{5^{24}}=\frac{27^{12}}{25^{12}}=\left(\frac{27}{25}\right)^{12}>1$ , so $3^{36}>5^{24}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $12$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 10 hard

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{50}$ , $3^{30}$ , $5^{20}$ , $26^{10}$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{50}$ , $3^{30}$ , $5^{20}$ , and $26^{10}$ , find the largest and the smallest.

Givens

The numbers $2^{50}$ , $3^{30}$ , $5^{20}$ , $26^{10}$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $50$ , $30$ , $20$ , and $10$ are all multiples of $10$ . Rewrite every number as something to the power $10$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

10

, so aim for the form

(\ )^{10}

.

50=5\times10,\ 30=3\times10,\ 20=2\times10

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{50}=(2^5)^{10}=32^{10},\ 3^{30}=(3^3)^{10}=27^{10},\ 5^{20}=(5^2)^{10}=25^{10}

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

25<26<27<32 \Rightarrow 25^{10}<26^{10}<27^{10}<32^{10}

Multiplying 10 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

10

with the original number.

5^{20}<26^{10}<3^{30}<2^{50}

32^{10}

is

2^{50}

, and so on.

Answer:

2^{50}

,

5^{20}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{50}\approx1.1\times10^{15}$ , $3^{30}\approx2.1\times10^{14}$ , $5^{20}\approx9.5\times10^{13}$ , $26^{10}\approx1.4\times10^{14}$ . The biggest is $2^{50}$ and the smallest is $5^{20}$ .

Another way: Compare pairs by dividing: for example $\frac{2^{50}}{3^{30}}=\frac{32^{10}}{27^{10}}=\left(\frac{32}{27}\right)^{10}>1$ , so $2^{50}>3^{30}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $10$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 11 hard

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{60}$ , $3^{40}$ , $7^{30}$ , $13^{20}$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{60}$ , $3^{40}$ , $7^{30}$ , and $13^{20}$ , find the largest and the smallest.

Givens

The numbers $2^{60}$ , $3^{40}$ , $7^{30}$ , $13^{20}$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases and the exponents are all different, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $60$ , $40$ , $30$ , and $20$ are all multiples of $10$ . Rewrite every number as something to the power $10$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

10

, so aim for the form

(\ )^{10}

.

60=6\times10,\ 40=4\times10,\ 30=3\times10,\ 20=2\times10

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{60}=(2^6)^{10}=64^{10},\ 3^{40}=(3^4)^{10}=81^{10},\ 7^{30}=(7^3)^{10}=343^{10},\ 13^{20}=(13^2)^{10}=169^{10}

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

64<81<169<343 \Rightarrow 64^{10}<81^{10}<169^{10}<343^{10}

Multiplying 10 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

10

with the original number.

2^{60}<3^{40}<13^{20}<7^{30}

64^{10}

is

2^{60}

, and so on.

Answer:

7^{30}

,

2^{60}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{60}\approx1.2\times10^{18}$ , $3^{40}\approx1.2\times10^{19}$ , $7^{30}\approx2.3\times10^{25}$ , $13^{20}\approx1.9\times10^{22}$ . The biggest is $7^{30}$ and the smallest is $2^{60}$ .

Another way: Compare pairs by dividing: for example $\frac{7^{30}}{13^{20}}=\frac{343^{10}}{169^{10}}=\left(\frac{343}{169}\right)^{10}>1$ , so $7^{30}>13^{20}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $10$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Variant 12 hard

Of the four numbers below, find the largest one and the smallest one, in that order.

$2^{60}$ , $3^{40}$ , $10^{20}$ , $7^{20}$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Among $2^{60}$ , $3^{40}$ , $10^{20}$ , and $7^{20}$ , find the largest and the smallest.

Givens

The numbers $2^{60}$ , $3^{40}$ , $10^{20}$ , $7^{20}$ .

Unknowns

The largest number and the smallest number.

Constraints

The bases are all different and the exponents are not all the same, so the numbers cannot be compared directly.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #15 Organize Information in More Ways

Instead of comparing bases with different exponents, change focus to the exponents: $60$ , $40$ , $20$ , and $20$ are all multiples of $20$ . Rewrite every number as something to the power $20$ , and then only the bases need comparing.

3 · Execute4 carry out the plan

1Pick a common exponent

#16 Count the Complement 8.EE.A.1

All four exponents are multiples of

20

, so aim for the form

(\ )^{20}

.

60=3\times20,\ 40=2\times20

A shared exponent turns four comparisons into comparing four bases.

2Rewrite each power

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

Use the power-of-a-power rule in reverse.

2^{60}=(2^3)^{20}=8^{20},\ 3^{40}=(3^2)^{20}=9^{20}

(a^m)^n=a^{mn}

, read backwards.

3Compare the bases

#15 Organize Information in More Ways 6.EE.A.1

With the same positive exponent, a bigger positive base gives a bigger power.

7<8<9<10 \Rightarrow 7^{20}<8^{20}<9^{20}<10^{20}

Multiplying 20 copies of a bigger number gives a bigger product.

4Translate back

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

Replace each base to the power

20

with the original number.

7^{20}<2^{60}<3^{40}<10^{20}

8^{20}

is

2^{60}

, and so on.

Answer:

10^{20}

,

7^{20}

4 · Reviewdoes it hold up?

Rough sizes agree: $2^{60}\approx1.2\times10^{18}$ , $3^{40}\approx1.2\times10^{19}$ , $10^{20}\approx1.0\times10^{20}$ , $7^{20}\approx8.0\times10^{16}$ . The biggest is $10^{20}$ and the smallest is $7^{20}$ .

Another way: Compare pairs by dividing: for example $\frac{10^{20}}{3^{40}}=\frac{10^{20}}{9^{20}}=\left(\frac{10}{9}\right)^{20}>1$ , so $10^{20}>3^{40}$ ; doing this for each pair gives the same order.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Rewriting each number as a power with exponent $20$.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Comparing whole-number powers with the same exponent.

💡Takeaway. Different bases and exponents? Make the exponents match -- then the bases tell you everything.

Match the exponents, then compare bases · 12 practice problems

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