 Power of a power with integer bases

Lesson
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?

Exploration

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 exponent 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 exponent 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 exponential form:

$\left(9^4\right)^3$(94)3

Question 2

Simplify using the exponent laws:

$\left(3^5\right)^3\times\left(3^2\right)^3$(35)3×(32)3

Question 3

Simplify, using the exponent laws:

$\left(2^3\right)^0\times\left(2^2\right)^3$(23)0×(22)3

Outcomes

9D.NA1.03

Derive, through the investigation and examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one and two variables with positive exponents