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.

We now need to consider what happens if we have a coefficient as well as an unknown raised to a negative power.

Examples

Question 1

Express $3x^{-3}$3x−3 with a positive index.

Think - the $x$x has been raised to a power of $-3$−3. We must evaluate this first, then multiply by $3$3.

Do - Apply the negative index law, then multiply by $3$3.

$3x^{-3}$3`x`−3	$=$=	$3\times x^{-3}$3×`x`−3
	$=$=	$3\times\frac{1}{x^3}$3×1`x`3
	$=$=	$\frac{3}{x^3}$3`x`3

As the power is only on the $x$x, the value of the coefficient hasn't changed.

A Fraction to a Negative Power

Things become a bit more complicated when we need to raise a fraction to a negative power.

First, you may want to remind yourself what we mean by a reciprocal.

The reciprocal of $\frac{1}{8}$18 is $8$8. The reciprocal of $\frac{2}{3}$23 is $\frac{3}{2}$32. The reciprocal of $91$91 is $\frac{1}{91}$191.

Can you see what reciprocal means? Simply, if you have a fraction, you need to invert it (or flip it) to find the reciprocal. If you have an integer, you put that integer as the denominator, and $1$1 as the numerator.

This links to negative powers, as we saw above.

Example

Question 2

So what happens if you are asked to calculate $\left(\frac{a}{b}\right)^{-3}$(ab)−3?

$\left(\frac{a}{b}\right)^{-3}$(`ab`)−3	$=$=	$\frac{a^{1\times\left(-3\right)}}{b^{1\times\left(-3\right)}}$`a`1×(−3)`b`1×(−3)	First we need to multiply out the powers.
	$=$=	$\frac{a^{-3}}{b^{-3}}$`a`−3`b`−3	We know that this is the same as
	$=$=	$a^{-3}\div b^{-3}$`a`−3÷`b`−3	Now let's rewrite this expression using only positive powers
	$=$=	$\frac{1}{a^3}\div\frac{1}{b^3}$1`a`3÷1`b`3	Dividing by a fraction is the same as multiplying by the reciprocal of that fraction
	$=$=	$\frac{1}{a^3}\times b^3$1`a`3×`b`3	Can you see what is going to happen next? By multiplying through we now get
	$=$=	$\frac{b^3}{a^3}$`b`3`a`3	Let's take a moment to compare this answer to the question.

$\left(\frac{a}{b}\right)^{-3}=\frac{b^3}{a^3}$(ab)−3=b3a3

Look for a shortcut here.

What has happened is we have found the reciprocal of the question, and raised each term to the power which is now a positive.

Question 3

Now let's try another.

Express $\left(\frac{x^8}{y^5}\right)^{-4}$(x8y5)−4 with a positive index.

	$=$=	$\frac{x^{8\times\left(-4\right)}}{y^{5\times\left(-4\right)}}$`x`8×(−4)`y`5×(−4)
	$=$=	$\frac{x^{-32}}{y^{-20}}$`x`−32`y`−20
	$=$=	$\frac{y^{20}}{x^{32}}$`y`20`x`32

Now it's your turn to practice applying these rules.

Worked Examples

Question 4

Express $4y^{-2}\times2y^{-4}$4y−2×2y−4 with a positive index.

Question 5

Express $\left(\frac{x^2}{y^4}\right)^{-1}$(x2y4)−1 without negative indices.

Index notation with variable bases and negative powers