Products and quotients with variable bases and negative indices

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:

$a^{-x}=\frac{1}{a^x}$a−x=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}$2y9×3y−5 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 $\left(4m^{-10}\right)^4$(4m−10)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 $m^{-10}$m−10 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

Question 4