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2.05 Negative indices

Lesson

Negative indices

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

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

Expanded form of A cubed divided by A to the power of 5. Ask your teacher for more information.

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.

So using the second approach, we can also express \dfrac{a^3}{ a^5} with a positive index as a^{-2}. This gives us the negative index law. When dealing with algebraic bases we follow exact the same approach.

For any base a, \, a^{-x} = \dfrac{1}{a^x}, x \neq 0. That is, when raising a base to a negative power:

  • Take the reciprocal of the expression

  • Turn the power into a positive

Examples

Example 1

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

Worked Solution
Create a strategy

Use the negative index law.

Apply the idea
\displaystyle \frac{1}{25}\displaystyle =\displaystyle \frac{1}{5^{2}}Write the denominator as a power of 5
\displaystyle =\displaystyle 5^{-2}Apply the negative index law
\displaystyle 5^n\displaystyle =\displaystyle 5^-2Equate the right hand side
\displaystyle n\displaystyle =\displaystyle -2Equate the powers

Example 2

Simplify the following, giving your answer with a positive index: \dfrac{9x^2}{3x^9}

Worked Solution
Create a strategy

Use the division index law: \dfrac{a^m}{a^n}=a^{m-n}

Apply the idea
\displaystyle \frac{9x^2}{3x^9}\displaystyle =\displaystyle 3x^{2-9}Subtract the powers and divide the coefficients
\displaystyle =\displaystyle 3x^{-7}Simplify the power
\displaystyle =\displaystyle \frac{3}{x^7}Apply the negative index law
Idea summary

For any base a,a^{-x} = \frac{1}{a^x}, x\neq 0

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

  • Take the reciprocal of the expression

  • Turn the power into a positive

Fractional bases with negative indices

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

To express \left(\dfrac{a}{b}\right)^{-3} with a positive index, we combine the rules for raising fractions to a power with the rule for negative indices: \left(\dfrac{a}{b}\right)^{n}=\dfrac{a^n}{b^n} and a^{-n}=\dfrac{1}{a^n}.

\displaystyle \left(\dfrac{a}{b}\right)^{-3}\displaystyle =\displaystyle \dfrac{a^{-3}}{b^{-3}}Apply the power to the fraction
\displaystyle =\displaystyle \dfrac{\dfrac{1}{a^{3}}}{\dfrac{1}{b^{3}}}Apply the negative index law
\displaystyle =\displaystyle \dfrac{1}{a^{3}}\times\dfrac{b^{3}}{1}Multiply by the reciprocal
\displaystyle =\displaystyle \frac{b^{3}}{a^{3}}Simplify the products
\displaystyle =\displaystyle \left(\dfrac{b}{a}\right)^{3}Write as a fraction to a power

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.

Examples

Example 3

Simplify the following, giving your answer with a positive index: \left(\dfrac{y}{4}\right)^{-3}

Worked Solution
Create a strategy

Find the reciprocal of the inside fraction and turn the power positive.

Apply the idea
\displaystyle \left(\frac{y}{4}\right)^{-3}\displaystyle =\displaystyle \left(\frac{4}{y}\right)^{3}Flip the fraction and turn the power positive
\displaystyle =\displaystyle \frac{4^3}{y^3}Apply the power to the fraction
\displaystyle =\displaystyle \frac{64}{y^3}Evaluate the powers
Idea summary

When raising a fraction to a negative power, we flip the fraction and change the power to positive: \left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}

Outcomes

MA5.2-6NA

simplifies algebraic fractions, and expands and factorises quadratic expressions

MA5.2-7NA

applies index laws to operate with algebraic expressions involving integer indices

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