Power of a power with integer bases
Exploration
| $\left(5^2\right)^3$(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.
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:
$\left(9^4\right)^3$(94)3
Question 2
Simplify using the index laws:
$\left(3^5\right)^3\times\left(3^2\right)^3$(35)3×(32)3
Question 3
Simplify, using the index laws:
$\left(2^3\right)^0\times\left(2^2\right)^3$(23)0×(22)3