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Study Guides > Intermediate Algebra

Read: Negative and Zero Exponent Rules

Learning Objectives

  • Define and use the zero exponent rule
  • Define and use the negative exponent rule

Define and use the zero exponent rule

Return to the quotient rule. We worked with expressions for which  a>ba>b so that the difference aba-b would never be zero or negative.

The Quotient (Division) Rule for Exponents

For any non-zero number x and any integers a and b: xaxb=xab \displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}
What would happen if a=ba=b? In this case, we would use the zero exponent rule of exponents to simplify the expression to 11. To see how this is done, let us begin with an example.

t8t8=t8t8=1\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1

If we were to simplify the original expression using the quotient rule, we would have
t8t8=t88=t0\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}
If we equate the two answers, the result is t0=1{t}^{0}=1. This is true for any nonzero real number, or any variable representing a real number.
a0=1{a}^{0}=1
The sole exception is the expression 00{0}^{0}. This appears later in more advanced courses, but for now, we will consider the value to be undefined.

The Zero Exponent Rule of Exponents

For any nonzero real number aa, the zero exponent rule of exponents states that
a0=1{a}^{0}=1

Example

Simplify each expression using the zero exponent rule of exponents.
  1. c3c3\Large\frac{{c}^{3}}{{c}^{3}}
  2. 3x5x5\Large\frac{-3{x}^{5}}{{x}^{5}}
  3. (j2k)4(j2k)(j2k)3\Large\frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}
  4. 5(rs2)2(rs2)2\Large\frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}

Answer: Use the zero exponent and other rules to simplify each expression.

  1. \Large\begin{array}\text{ }\frac{c^{3}}{c^{3}} \hfill& =c^{3-3} \\ \hfill& =c^{0} \\ \hfill& =1\end{array}
  2. 3x5x5=3x5x5=3x55=3x0=31=3\Large\begin{array}{ccc}\hfill \frac{-3{x}^{5}}{{x}^{5}}& =& -3\cdot \frac{{x}^{5}}{{x}^{5}}\hfill \\ & =& -3\cdot {x}^{5 - 5}\hfill \\ & =& -3\cdot {x}^{0}\hfill \\ & =& -3\cdot 1\hfill \\ & =& -3\hfill \end{array}
  3. (j2k)4(j2k)(j2k)3=(j2k)4(j2k)1+3Use the product rule in the denominator.=(j2k)4(j2k)4Simplify.=(j2k)44Use the quotient rule.=(j2k)0Simplify.=1\large\begin{array}{cccc}\hfill \frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}& =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{1+3}}\hfill &\text{Use the product rule in the denominator}.\hfill \\ & =&\frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}}\hfill & \text{Simplify}.\hfill \\ & =&{\left({j}^{2}k\right)}^{4 - 4}\hfill &\text{Use the quotient rule}.\hfill \\ & =& {\left({j}^{2}k\right)}^{0}\hfill&\text{Simplify}.\hfill \\ & =& 1& \end{array}
  4. 5(rs2)2(rs2)2=5(rs2)22Use the quotient rule.=5(rs2)0Simplify.=51Use the zero exponent rule.=5Simplify.\large\begin{array}{cccc}\hfill \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& =&5{\left(r{s}^{2}\right)}^{2 - 2}\hfill &\text{Use the quotient rule}.\hfill \\ & =& 5{\left(r{s}^{2}\right)}^{0}\hfill &\text{Simplify}.\hfill \\ & =& 5\cdot 1\hfill &\text{Use the zero exponent rule}.\hfill \\ & =& 5\hfill &\text{Simplify}.\hfill \end{array}

In the following video you will see more examples of simplifying expressions whose exponents may be zero. https://youtu.be/rpoUg32utlc  

Define and use the negative exponent rule

Another useful result occurs if we relax the condition that a>ba>b in the quotient rule even further. For example, can we simplify h3h5\frac{{h}^{3}}{{h}^{5}}? When a<ba<b—that is, where the difference aba-b is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal. Divide one exponential expression by another with a larger exponent. Use our example, h3h5\frac{{h}^{3}}{{h}^{5}}.
h3h5=hhhhhhhh=hhhhhhhh=1hh=1h2\Large\begin{array}{ccc}\hfill \frac{{h}^{3}}{{h}^{5}}& =& \frac{h\cdot h\cdot h}{h\cdot h\cdot h\cdot h\cdot h}\hfill \\ & =& \frac{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}}{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}\cdot h\cdot h}\hfill \\ & =& \frac{1}{h\cdot h}\hfill \\ & =& \frac{1}{{h}^{2}}\hfill \end{array}
If we were to simplify the original expression using the quotient rule, we would have
h3h5=h35= h2\Large\begin{array}{ccc}\hfill \frac{{h}^{3}}{{h}^{5}}& =& {h}^{3 - 5}\hfill \\ & =& \text{ }{h}^{-2}\hfill \end{array}
Putting the answers together, we have h2=1h2{h}^{-2}=\frac{1}{{h}^{2}}. This is true for any nonzero real number, or any variable representing a nonzero real number. A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
an=1anandan=1an\begin{array}{ccc}{a}^{-n}=\frac{1}{{a}^{n}}& \text{and}& {a}^{n}=\frac{1}{{a}^{-n}}\end{array}
We have shown that the exponential expression an{a}^{n} is defined when nn is a natural number, 00, or the negative of a natural number. That means that an{a}^{n} is defined for any integer nn. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer nn.  

The Negative Rule of Exponents

For any nonzero real number aa and natural number nn, the negative rule of exponents states that
an=1an{a}^{-n}=\frac{1}{{a}^{n}}
 

Example

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.
  1. (2b)3(2b)10\Large\frac{{(2b) }^{3}}{{(2b) }^{10}}
  2. z2zz4\Large\frac{{z}^{2}\cdot z}{{z}^{4}}
  3. (5t3)4(5t3)8\Large\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}

Answer:

Solution

  1. (2b)3(2b)10=(2b)310=(2b)7=1(2b)7\Large\frac{{(2b) }^{3}}{{(2b)}^{10}}={(2b)}^{3 - 10}={(2b) }^{-7}=\frac{1}{{(2b)}^{7}}
  2. z2zz4=z2+1z4=z3z4=z34=z1=1z\Large\frac{{z}^{2}\cdot z}{{z}^{4}}=\frac{{z}^{2+1}}{{z}^{4}}=\frac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\frac{1}{z}
  3. (5t3)4(5t3)8=(5t3)48=(5t3)4=1(5t3)4\Large\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}={\left(-5{t}^{3}\right)}^{4 - 8}={\left(-5{t}^{3}\right)}^{-4}=\frac{1}{{\left(-5{t}^{3}\right)}^{4}}

In the following video you will see examples of simplifying expressions with negative exponents. https://youtu.be/Gssi4dBtAEI

Combine exponent rules to simplify expressions

In the next examples we will combine the use of the product and quotient rules to simplify expressions whose terms may have negative or zero exponents.

Example

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.
  1. b2b8{b}^{2}\cdot {b}^{-8}
  2. (x)5(x)5{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}
  3. 7z(7z)5\frac{-7z}{{\left(-7z\right)}^{5}}

Answer:

Solution

  1. b2b8=b28=b6=1b6{b}^{2}\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\frac{1}{{b}^{6}}
  2. (x)5(x)5=(x)55=(x)0=1{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5 - 5}={\left(-x\right)}^{0}=1
  3. 7z(7z)5=(7z)1(7z)5=(7z)15=(7z)4=1(7z)4\frac{-7z}{{\left(-7z\right)}^{5}}=\frac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1 - 5}={\left(-7z\right)}^{-4}=\frac{1}{{\left(-7z\right)}^{4}}

The following video shows more examples of how to combine the use of the product and quotient rules to simplify expressions whose terms may have negative or zero exponents. https://youtu.be/EkvN5tcp4Cc

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