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

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

Question 3

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