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5. Exponents, Radicals, & Exponential Functions
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Algebra 1

5.04 Zero and negative exponents

Lesson
Practice

Zero and negative exponents

Now let's look at the quotient of powers rule when m and n are equal.

Exploration

Drag the slider to fold the paper in half. Explore the number of sections created on the paper after any number of folds.

Alternatively, take a sheet of paper and fold it in half, unfolding it each time to count the number of sections created.

Loading interactive...

  1. Use the applet to complete the table of values for the number of sections in the paper created based on the number of folds we make:

    Number of folds (power)2^02^12^22^32^4
    Number of sections created

  2. As the number of folds decreases, what is the pattern in the number of sections created?

  3. Moving from right to left, complete the table of values using the pattern you found above:

    4^{-2}4^{-1}4^04^14^24^3
    1664

The zero rule states that a^{0} = 1.

The zero rule tells us any non-zero base raised to the power of zero is equal to 1.

For example:

\displaystyle a^0\displaystyle =\displaystyle a^{3-3}Substitute 0=3-3
\displaystyle =\displaystyle \dfrac{a^3}{a^3}Quotient rule
\displaystyle =\displaystyle \dfrac{a\cdot a\cdot a}{a\cdot a\cdot a}Write in expanded form
\displaystyle =\displaystyle \dfrac{\cancel{a}\cdot \cancel{a}\cdot \cancel{a}}{\cancel{a}\cdot \cancel{a}\cdot \cancel{a}}Divide out common factors
\displaystyle a^0\displaystyle =\displaystyle 1

The negative exponent rule states: a^{-x}=\dfrac{1}{a^{x}}

The negative exponent rule tells us any non-zero base raised to a negative exponent is equal to 1 divided by the same base raised to the opposite positive exponent. Consider the following:

\displaystyle \dfrac{a^2}{a^5}\displaystyle =\displaystyle \dfrac{a \cdot a}{a\cdot a \cdot a \cdot a \cdot a} Write in expanded form
\displaystyle =\displaystyle \dfrac{a \cdot a}{a\cdot a } \cdot \dfrac{1}{a \cdot a \cdot a} Group common factors
\displaystyle =\displaystyle 1 \cdot \dfrac{1}{a \cdot a \cdot a} Divide common factors
\displaystyle =\displaystyle \dfrac{1}{a \cdot a \cdot a} Multiplicative identity
\displaystyle \dfrac{a^2}{a^5}\displaystyle =\displaystyle \dfrac{1}{a^3} Write in exponential form

And also:

\displaystyle \dfrac{a^2}{a^5}\displaystyle =\displaystyle a^{2-5}Quotient rule
\displaystyle =\displaystyle a^{-3}Evaluate the subtraction

Therefore, a^{-3}=\dfrac{a^2}{a^5}=\dfrac{1}{a^3}.

We can use this rule to write negative exponents as positive exponents or positive exponents as negative exponents.

Examples

Example 1

Simplify x^{5}\div x^{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 x^{5}\div x^{5}\displaystyle =\displaystyle \dfrac{x^{5}}{x^{5}}Write the expression as fraction
\displaystyle =\displaystyle \dfrac{x \cdot x\cdot x\cdot x\cdot x}{x \cdot x\cdot x\cdot x\cdot x}Write in expanded form
\displaystyle =\displaystyle \dfrac{x}{x}Divide out common factors
\displaystyle =\displaystyle 1Simplify
Reflect and check

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

This should be no surprise; the initial expression asks us "What do we get when we divide x^{5} by itself?", to which the answer is simply1, since anything divided by itself is equivalent to 1.

Example 2

Simplify 9p^{0}.

Worked Solution
Create a strategy

We can use the zero rule: a^{0}=1

Apply the idea

Since the base of p has a power of 0, the whole expression is equal to 9 \cdot 1. So by simplifying this, we have:

9p^{0} = 9 \cdot 1 =9

Example 3

Write the following with a negative exponent: \dfrac{1}{g} \cdot \dfrac{1}{g} \cdot \dfrac{1}{g} \cdot \dfrac{1}{g}

Worked Solution
Create a strategy

We can use the product rule, a^{m} \cdot a^{n}=a^{m+n}, and the negative exponent rule, a^{-x}=\dfrac{1}{a^{x}}.

Apply the idea

Each fraction is to the power of 1 so we can start by applying the product rule.

\displaystyle \dfrac{1}{g} \cdot \dfrac{1}{g} \cdot \dfrac{1}{g} \cdot \dfrac{1}{g}\displaystyle =\displaystyle \left(\dfrac{1}{g}\right)^{(1+1+1+1)}Apply the product rule
\displaystyle =\displaystyle \left(\dfrac{1}{g}\right)^{4}Evaluate the addition
\displaystyle =\displaystyle \left(g^{-1}\right)^4Negative exponent rule
\displaystyle =\displaystyle g^{-4}Power rule

Therefore, \dfrac{1}{g} \cdot \dfrac{1}{g} \cdot \dfrac{1}{g} \cdot \dfrac{1}{g} expressed with a negative exponent is g^{-4}.

Example 4

Express the following with positive exponents. 3a^{2}\cdot x^{-4}\cdot 5\cdot x^{-2} \cdot a^{6}

Worked Solution
Create a strategy

We can use product of powers rule: a^{m} \cdot a^{n}=a^{m+n}

Apply the idea
\displaystyle 3a^{2}\cdot x^{-4}\cdot 5 \cdot x^{-2} \cdot a^{6}\displaystyle =\displaystyle 15a^{2}\cdot x^{-4}\cdot x^{-2} \cdot a^{6}Multiply the coefficients
\displaystyle =\displaystyle 15a^{2}\cdot a^{6}\cdot x^{-4}\cdot x^{-2} Commutative property
\displaystyle =\displaystyle 15a^{2+6}\cdot x^{-4+\left(-2\right)}Product rule
\displaystyle =\displaystyle 15\cdot a^{8}\cdot x^{-6}Evaluate the addition
\displaystyle =\displaystyle 15\cdot a^{8}\cdot \dfrac{1}{x^{6}}Negative exponent rule
\displaystyle =\displaystyle \dfrac{15a^{8}}{x^{6}}Multiply
Idea summary

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

The negative exponent rule states: a^{-x}=\dfrac{1}{a^{x}}

Raise a fraction to a negative exponent

We can use this same understanding to raise a fraction to a negative exponent.

Let's first review what a reciprocal is.

The reciprocal of \dfrac{1}{8} is \dfrac{8}{1}=8. The reciprocal of \dfrac{2}{3} is \dfrac{3}{2}. The reciprocal of 91 is \dfrac{1}{91}.

So to find the reciprocal, you need to invert or flip the fraction. If you have an integer, you put that integer as the denominator, and 1 as the numerator.

Exploration

Consider the expression \left(\dfrac{a}{b} \right)^{-1}. We want to write this without negative exponents.

\displaystyle \left(\dfrac{a}{b} \right)^{-1}\displaystyle =\displaystyle \dfrac{⬚}{⬚}Apply the power to the terms inside the parentheses
\displaystyle =\displaystyle ⬚ \div ⬚Rewrite as a division
\displaystyle =\displaystyle ⬚ \div ⬚Apply the negative exponent rule
\displaystyle =\displaystyle ⬚ \cdot ⬚Multiply by the reciprocal
\displaystyle =\displaystyle ⬚Simplify

  1. Complete the working above by filling in the blanks.

  2. Complete the statement: In general, \left(\dfrac{a}{b}\right)^{-1}=⬚

When raising a fraction to any negative exponent, we find the reciprocal of the fraction in the parentheses, then apply the quotient of powers rule: \left(\dfrac{a}{b} \right)^{-n}=\left(\dfrac{b}{a}\right)^n=\dfrac{b^n}{a^n}

Examples

Example 5

Express \left(\dfrac{y^{2}}{z^{4}}\right)^{-1} using positive exponents.

Worked Solution
Create a strategy

We can use the exponential rule: \left(\dfrac{a}{b}\right)^{-1} = \dfrac{b}{a}

Apply the idea
\displaystyle \left(\dfrac{y^{2}}{z^{4}}\right)^{-1}\displaystyle =\displaystyle \dfrac{z^{4}}{y^{2}}Rewrite using the reciprocal

Example 6

Write \left(\dfrac{3x^{2}}{2y^{3}}\right)^{-2} with positive exponents.

Worked Solution
Create a strategy

First rewrite with a positive exponent and then apply the quotient of powers rule.

Apply the idea
\displaystyle \left(\dfrac{3x^{2}}{2y^{3}}\right)^{-2}\displaystyle =\displaystyle \left(\dfrac{2y^{3}}{3x^{2}}\right)^{2}Apply the negative exponent rule
\displaystyle =\displaystyle \dfrac{\left(2)^2(y^{3}\right)^2}{\left(3)^2(x^{2}\right)^2}Apply the quotient of powers rule
\displaystyle =\displaystyle \dfrac{4y^6}{9x^4}Apply the power rule
Reflect and check

An alternative method for solving this problem is applying the quotient of powers rule first, then the power rule, finally the negative exponent rule:

\displaystyle \left(\dfrac{3x^{2}}{2y^{3}}\right)^{-2}\displaystyle =\displaystyle \dfrac{\left(3)^{-2}(x^2\right)^{-2}}{\left(2)^{-2}(y^3\right)^{-2}}Apply the quotient of powers rule
\displaystyle =\displaystyle \dfrac{3^{-2}x^{-4}}{2^{-2}y^{-6}}Apply the power rule
\displaystyle =\displaystyle 3^{-2}x^{-4}\div 2^{-2}y^{-6}Write as a division
\displaystyle =\displaystyle \dfrac{1}{3^2x^4}\div \dfrac{1}{2^2 y^6}Negative exponent rule
\displaystyle =\displaystyle \dfrac{1}{9x^4}\cdot \dfrac{4y^6}{1}Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{4y^6}{9x^4}Multiply

Example 7

Express the following with a positive exponent: \dfrac{c^{3}}{c^{4}\cdot d^{-4}}

Worked Solution
Create a strategy

We can use the quotient of powers rule and power rule: a^{m} \div a^{n}=\dfrac{a^{m}}{a^{n}}=a^{m-n}

Apply the idea
\displaystyle \dfrac{c^{3}}{c^{4}\cdot d^{-4}}\displaystyle =\displaystyle \dfrac{c^{(3-4)}}{d^{-4}}Quotient rule
\displaystyle =\displaystyle \dfrac{c^{-1}}{d^{-4}}Evaluate the subtraction
\displaystyle =\displaystyle \dfrac{d^{4}}{c}Negative exponent rule
Idea summary

When raising a fraction to a negative exponent, we get:

\left(\dfrac{a}{b} \right)^{-1}=\dfrac{b}{a}and \left(\dfrac{a}{b} \right)^{-n}=\left(\dfrac{b}{a}\right)^{n}=\dfrac{b^{n}}{a^{n}}

Outcomes

A.EO.3

The student will derive and apply the laws of exponents.

A.EO.3a

Derive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases.

A.EO.3b

Simplify multivariable expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.

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