Indices

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?

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.

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,

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

Express in simplified index form:

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

Simplify using the index laws:

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

Simplify, using the index laws:

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