We already learnt to apply the power of a power rule to positive indices. This rule states:
$\left(x^a\right)^b=x^{a\times b}$(xa)b=xa×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:
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:
The negative index law states:
$a^{-x}=\frac{1}{a^x}$a−x=1ax
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: $\left(w^{10}\right)^{-5}$(w10)−5.
Think: Using the power of a power rule, we need to multiply the powers.
$\left(w^{10}\right)^{-5}=w^{10\times\left(-5\right)}$(w10)−5=w10×(−5)
Do: $\left(w^{10}\right)^{-5}=w^{-50}$(w10)−5=w−50
Express $\left(5y^3\right)^{-3}$(5y3)−3 with a positive index.
Simplify the following:
$\left(-4u^{-4}\right)^3$(−4u−4)3