Topics
2. Exponents & Scientific Notation
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United States of America
Grade 8

2.07 Power of a division and zero exponent properties

Lesson
Practice

Introduction

We have previously looked at the quotient property, which is: \dfrac{a^{m}}{a^{n}}=a^{m-n}, where a is any base number, and m and n are powers.

We'll now consider two related scenarios. Firstly, we'll look at what happens when we take a power of a fraction, such as \left(\dfrac{2}{4}\right)^{2}. Secondly, we'll consider what happens when we are using the quotient property and m and n are equal, for example \dfrac{2^{5}}{2^{5}}.

Power of a division property

Consider the expression \left(\dfrac{2}{4}\right)^2. Let's look at how we would rewrite this in expanded form:

\displaystyle \left(\dfrac{2}{4}\right)^{2}\displaystyle =\displaystyle \dfrac{2}{4}\times\dfrac{2}{4}Expanding the fraction
\displaystyle =\displaystyle \dfrac{2\times 2}{4\times 4}
\displaystyle =\displaystyle \dfrac{2^{2}}{4^{2}}

As you can see, the power being applied to the whole fraction is now being applied to the top and bottom terms of the fraction. This is what the power of a division property states:\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}

Examples

Example 1

Simplify the following: \left(\dfrac{4}{5}\right)^{3}

Worked Solution
Create a strategy

We can use the power of a division property: \left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}

Apply the idea
\displaystyle \left(\dfrac{4}{5}\right)^{3}\displaystyle =\displaystyle \dfrac{4^{3}}{5^{3}}Power of a division property
\displaystyle =\displaystyle \dfrac{4\times4\times4}{5\times5\times5}Expanded form
\displaystyle =\displaystyle \dfrac{64}{125}

64 and 125 share no common factors so the fraction is fully simplfied.

Idea summary

When we find a fraction to a power, the power can be applied to the top and bottom terms of the fraction individually: \left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}

Zero power property

Now let's look at the quotient property when m and n are equal. We'll also use the the zero power property, which states that a^{0} = 1.

Examples

Example 2

Simplify 2^{5}\div 2^{5} by first writing the expression in expanded form.

Worked Solution
Create a strategy

Write the expression as fraction and write it in expanded form to cancel out the common factors.

Apply the idea
\displaystyle 2^{5}\div 2^{5}\displaystyle =\displaystyle \dfrac{2^{5}}{2^{5}}Write the expression as fraction
\displaystyle =\displaystyle \dfrac{2 \times 2\times 2\times 2\times 2}{2 \times 2\times 2\times 2\times 2}Write the fraction in expanded form
\displaystyle =\displaystyle \dfrac{1}{1}Cancel out each similar factor
\displaystyle =\displaystyle 1Simplify
Reflect and check

Recall that a^{m}\div a^{n}= a^{m-n}, so we could have written 2^{5}\div 2^{5} as 2^{5-5}=2^0. Now we can see that 2^{0}=1.

This should be no surprise; the initial expression asks us "What do we get when we divide 2^{5} by itself?", to which the answer is simply 1, since anything divided by itself is equivalent to 1.

Example 3

Simplify (6)^{0}.

Worked Solution
Create a strategy

We can use the zero power property: a^{0}=1.

Apply the idea

Since the base of 6 has a power of 0, the whole expression is equal to 1.

(6)^{0} = 1

Idea summary

For any numeric or algebraic expression a, the zero power property tells us that a^{0}=1

Outcomes

8.EE.A.1

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

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