Hong Kong
Stage 1 - Stage 3

# Products and quotients with variable bases and negative indices

Lesson

We've already learnt about the multiplication and division laws and we've see that sometimes we get answers with negative indices. If you remember, expressions with negative indices can be expressed as their reciprocals with positive indices. The negative index law states:

$a^{-x}=\frac{1}{a^x}$ax=1ax

or if it is a fraction:

$\left(\frac{a}{b}\right)^{-x}=\left(\frac{b}{a}\right)^x$(ab)x=(ba)x

To answer these kinds of questions, we can multiply or divide the numbers (as the question states), then multiply or divide terms with like bases using the index laws. Click the links if you need a refresher on how to multiply or divide fractions.

#### Examples

##### Question 1

Express $2y^9\times3y^{-5}$2y9×3y5 with a positive index.

Think: We need to multiply the numbers, then apply the index multiplication law.

Do:

 $2y^9\times3y^{-5}$2y9×3y−5 $=$= $6y^{9+\left(-5\right)}$6y9+(−5) $=$= $6y^4$6y4

##### Question 2

Simplify $\left(4m^{-10}\right)^4$(4m10)4, expressing your answer in positive index form.

Think: We're going to use the power of a power rule, then the negative index rule. Remember both $4$4 and $m^{-10}$m10 are to the power of $4$4.

Do:

 $\left(4m^{-10}\right)^4$(4m−10)4 $=$= $4^4\times m^{-10\times4}$44×m−10×4 $=$= $256m^{-40}$256m−40 $=$= $\frac{256}{m^{40}}$256m40​

##### Question 3

Express $p^{-2}q^3$p2q3 as a fraction without negative indices.

##### Question 4

Express $\frac{25x^{-7}}{5x^{-4}}$25x75x4 with a positive index.