New Zealand
Level 7 - NCEA Level 2

# Fractional bases with fractional powers

Lesson

In Powerful Fractions, we learnt about the fractional index rule. Then, in Working with Powerful Fractions we learnt to apply this rule to terms with numerical bases. Now we are going to learn how to apply this rule to terms with fractional bases.

It's basically the same process that we've already learnt. We just need to remember that we need to apply the fractional index to both the numerator and the denominator.

The Fractional Index Rule with a Fractional Base

$\left(\frac{x}{y}\right)^{\frac{m}{n}}=\frac{x^{\frac{m}{n}}}{y^{\frac{m}{n}}}$(xy)mn=xmnymn

We could also write this as $\frac{\sqrt[n]{x^m}}{\sqrt[n]{y^m}}$nxmnym

Don't worry if this formula looks scary with all its algebraic terms. We'll run through some examples now so you can see it in action!

#### Examples

##### Question 1

Evaluate $\left(\frac{25}{36}\right)^{\frac{1}{2}}$(2536)12.

##### Question 2

Evaluate by first expressing as a surd.

$\left(\frac{8}{125}\right)^{\frac{2}{3}}$(8125)23

### Outcomes

#### M7-6

Manipulate rational, exponential, and logarithmic algebraic expressions

#### 91261

Apply algebraic methods in solving problems