# 3.10 Negative exponents

Lesson

## Negative exponents

We have seen how we can rewrite expressions with negative powers to have a positive powers

For example, if we simplified $a^3\div a^5$a3÷​a5 using the division law, we would get $a^{-2}$a2.  Let's expand the example to see why this is the case:

Remember that when we are simplifying fractions, we are looking to cancel out common factors in the numerator and denominator. Remember that any number divided by itself is $1$1.

So using the second approach, we can also express $a^3\div a^5$a3÷​a5 with a positive exponent as $\frac{1}{a^2}$1a2. This gives us the negative exponent law. When dealing with algebraic bases we follow exact the same approach.

Negative exponent law

For any base $a$a,

$a^{-x}=\frac{1}{a^x}$ax=1ax$x\ne0$x0.

That is, when raising a base to a negative power:

• Take the reciprocal of the expression
• Turn the power into a negative

## Fractional bases with negative exponents

When raising a fractional base to a negative power we can combine the individual rules we have seen.

#### Worked example

Express the following with a positive exponent: $\left(\frac{a}{b}\right)^{-3}$(ab)3

Think: We want to combine the rules for raising fractions with the rule for negative exponents.

That is we want to use the rules $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$(ab)n=anbn and $a^{-n}=\frac{1}{a^n}$an=1an.

Do:

 $\left(\frac{a}{b}\right)^{-3}$(ab​)−3 $=$= $\frac{a^{-3}}{b^{-3}}$a−3b−3​ Use the rule for raising a fraction $=$= $a^{-3}\div b^{-3}$a−3÷​b−3 Rewrite the quotient with a division symbol $=$= $\frac{1}{a^3}\div\frac{1}{b^3}$1a3​÷​1b3​ Apply the negative exponent rule to the numerator and the denominator to express both with positive exponents $=$= $\frac{1}{a^3}\times\frac{b^3}{1}$1a3​×b31​ Dividing by a fraction is the same as multiplying by the reciprocal of that fraction $=$= $\frac{b^3}{a^3}$b3a3​ Simplify the fractional product $=$= $\left(\frac{b}{a}\right)^3$(ba​)3 Write as a single term raised to a power by using the reverse of the rule for raising fractions

We can see that $\left(\frac{a}{b}\right)^{-3}=\frac{b^3}{a^3}$(ab)3=b3a3$=$=$\left(\frac{b}{a}\right)^3$(ba)3

Reflect:  What has happened is we have found the reciprocal of the expression in the question, and turned the power into a positive. Using this trick will save a lot of time!

Raising a fraction to a power

For any base number of the form $\frac{a}{b}$ab, and any number $n$n as a power,

$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$(ab)n=anbn

If $n$n is negative, then we also use the fact $a^{-n}=\frac{1}{a}$an=1a. Giving us the following rule:

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

#### Practice questions

##### Question 1

Find the value of $n$n such that $\frac{1}{25}=5^n$125=5n.

##### Question 2

Simplify the following, giving your answer with a positive exponent:

$\frac{9x^2}{3x^9}$9x23x9

##### Question 3

Simplify the following, giving your answer with a positive exponent:

$\left(\frac{y}{4}\right)^{-3}$(y4)3

### Outcomes

#### 9.B2.2

Analyse, through the use of patterning, the relationships between the exponents of powers and the operations with powers, and use these relationships to simplify numeric and algebraic expressions.

#### 9.C1.4

Simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools, in different contexts.