Products and quotients with variable bases and negative exponents

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

Think: We need to multiply the numbers, then apply the exponent 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 exponential form.

Think: We're going to use the power of a power rule, then the negative exponent 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