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Power of a power with integer bases

We previously looked at expressions like $7^2\times7^3=7^{2+3}$72×73=72+3, but how should we deal with a power of a power?


Consider the expression $\left(5^2\right)^3$(52)3. What is the resulting power of base $5$5? To find out, have a look at the expanded form of the expression:
$\left(5^2\right)^3$(52)3 $=$= $\left(5^2\right)\times\left(5^2\right)\times\left(5^2\right)$(52)×(52)×(52)
  $=$= $\left(5\times5\right)\times\left(5\times5\right)\times\left(5\times5\right)$(5×5)×(5×5)×(5×5)
  $=$= $5\times5\times5\times5\times5\times5$5×5×5×5×5×5
  $=$= $5^6$56

In the expanded form, we can see that we are multiplying six groups of $5$5 together. That is, $\left(5^2\right)^3=5^6$(52)3=56.

We can confirm this result using the index law of multiplication:

We know $\left(5^2\right)\times\left(5^2\right)\times\left(5^2\right)=5^{2+2+2}$(52)×(52)×(52)=52+2+2 which is equal to $5^6$56.


The index law for further powers

We can avoid having to write each expression in expanded form by using the power of a power law.

The power of a power law

For any base number $a$a, and any numbers $m$m and $n$n as powers,

$\left(a^m\right)^n=a^{m\times n}$(am)n=am×n

That is, when simplifying a term with a power that itself has a power:

  • Keep the same base
  • Find the product of the powers


Practice questions

Question 1

Express in simplified index form:


Question 2

Simplify using the index laws:


Question 3

Simplify, using the index laws:


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