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Glacius
FoundationsConcept reference

A power raised to a power

An integer power of an integer power of a nonzero real base equals that base raised to the product of the two exponents.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. In (2^3)^2, the outer exponent repeats the whole group 2^3 twice
  3. For a nonzero base the rule also holds for integer exponents
  4. Check your understanding
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01In (2^3)^2, the outer exponent repeats the whole group 2^3 twice#

In (23)2(2^3)^2, the outer exponent repeats the whole group 232^3 twice. Two groups of three factors make six.

Two groups of three factors of 2 make six factors: (2³)² = 2⁶.Two groups of three factors of 2 make six factors: (2³)² = 2⁶.
Figure 1Two groups of three factors of 2 make six factors: (2³)² = 2⁶.
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Multiply the factors per group by the number of groups:

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

For example, (32)3(3^2)^3 has three groups of two factors. Its value is 36=7293^6=729. The base remains 3.

Check your reasoning

Write (52)4=5k(5^2)^4=5^k. What is kk?

Show answer and explanation
8

Four groups of two factors give 2⋅4=82\cdot4=8 factors.

02For a nonzero base the rule also holds for integer exponents#

For a nonzero base the rule also holds for integer exponents. A reciprocal example is (23)−2=2−6=1/64(2^3)^{-2}=2^{-6}=1/64. Keep both exponents in the multiplication.

Check your reasoning

Someone writes (42)3=45(4^2)^3=4^5. Which exponent repairs it?

  1. A66
  2. B55
  3. C88
Show answer and explanation
66

A power of a power multiplies exponents: 2⋅3=62\cdot3=6.

03Check your understanding#

Check your reasoning

Each block is a product of four factors of 2. Multiply three identical blocks. The result is 2k2^k. What is kk?

Show answer and explanation
12

Three groups of four give 4⋅3=124\cdot3=12 factors.

Key takeaway

Count factors per group times the number of groups.

  • Rewrite a power raised to an integer power as one power for a nonzero base.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “A power raised to a power.” Math behind ML. /learn/f-power-of-power