Indices

NZ Level 7 (NZC) Level 2 (NCEA)

Power of a power with variable bases and negative powers

Lesson

We already learnt to apply the power of a power rule to positive indices. This rule states:

$\left(x^a\right)^b=x^{a\times b}$(`x``a`)`b`=`x``a`×`b`

Now we are going to explore what happens when we also include negative values in these kinds of questions.

If you think back to learning about multiplying and dividing by negative numbers, you'll remember that:

- Multiplying a
**positive**and a**negative**value will give you a**negative**answer. - Multiplying
**two negative**numbers together gives you a**positive**answer.

Let's think for a minute about what happens if we multiply a negative value by itself more than twice.

Say we had the question $\left(-2\right)^3$(−2)3. This means $-2\times\left(-2\right)\times\left(-2\right)$−2×(−2)×(−2). If we simplify this, $-2\times\left(-2\right)=4$−2×(−2)=4 and $-4\times\left(-2\right)=-8$−4×(−2)=−8.

What about if we multiplied it by itself again to get the answer to $\left(-2\right)^4$(−2)4? We know that $\left(-2\right)^3=-8$(−2)3=−8 and $-8\times\left(-2\right)=16$−8×(−2)=16.

So, as a general rule:

- If you raise a
**negative number**by an**even index**, you will get a**positive answer** - If you raise a
**negative number**by an**odd index**, you will get a**negative answer**

Remember

The negative index law states:

$a^{-x}=\frac{1}{a^x}$`a`−`x`=1`a``x`

So if you need to express a negative index as a positive index, or a positive index as a negative index, you need to convert it to a fraction using this rule.

**Simplify:** $\left(w^{10}\right)^{-5}$(`w`10)−5.

**Think:** Using the power of a power rule, we need to multiply the powers.

$\left(w^{10}\right)^{-5}=w^{10\times\left(-5\right)}$(`w`10)−5=`w`10×(−5)

**Do:** $\left(w^{10}\right)^{-5}=w^{-50}$(`w`10)−5=`w`−50

Express $\left(5y^3\right)^{-3}$(5`y`3)−3 with a positive index.

Simplify the following:

$\left(-4u^{-4}\right)^3$(−4`u`−4)3

Manipulate rational, exponential, and logarithmic algebraic expressions

Apply algebraic methods in solving problems