2.4 Exponent Laws: Part 1

Exponents can be used as a way to express repeated multiplication. For example, 3 \times 3 \times 3 \times 3 can be written in exponential form as 34 because 3 is being multiplied by itself four times.

A number written in exponential form takes the form an, where
a is the base,
n is the exponent, and
an is the power.

A coefficient is a number that multiplies a variable. For example, in the expression 2x5, the coefficient is 2.

Practice 1

Fill in the blanks.

a) The power 103 has a base of                  and an exponent of                .

b) The power (–5)2 has a base of                  and an exponent of                .

c) The power \left(\frac{4}{5}\right)^5 has a base of                  and an exponent of                .
Practice 2

Fill in the blanks.

a) The expression 6y3 has a coefficient of                .

b) The expression –8m7 has a coefficient of                .
Practice 3

Evaluate.

a) 73 b) (–3)2 c) –32 d) \left(-\frac{2}{3}\right)^4 e) (0.25)3

Exponents Laws

Exponent laws can be used to simplify and evaluate expressions. Using exponent laws can help reduce the number of steps needed to complete a problem.

Table 2.4.1
Name Rule Example
Product Law am \bullet an = am + n 23 \bullet 22 = 23 + 2 = 25 = 32
Quotient Law \frac{a^m}{a^n} = am–n
 \frac{y^4}{y^2} = y4–2 = y2
Power of Power Law (am)n = amn (x3)2 = x3 \cdot 2 = x6
Power of Product Law (ab)n = an \bullet bn (2 \bullet 3 )2 = 22 \bullet 32 = 4 \bullet 9 = 36
Power of Quotient Law \left(\frac{a}{b}\right)^n = \left(\frac{a^n}{b^n}\right) \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}
Zero Exponent Law a0 = 1 150 = 1
One Exponent Law a1 = a 71 = 7
Practice 4

Use the exponent laws to simplify. If possible, evaluate.

a) m2m4

 

 

 b) c3 \bullet c \bullet c0

 

 

 c) (–2)4(–2)3

 

 

 d) \frac{y^7}{y^5} 

 

   

 e) \frac{3^9}{3^7}   

 

 
Practice 5

Use the exponent laws to simplify. If possible, evaluate.

a) (b3)5

 

 b) (n7)2

 

 c) (42)3   

 

 d) (xy)4

 
e) (2mn)3 

 

 

  f) (–5x)2

 

 

 g) \left(\frac{x}{2}\right)^4   

 

 

 h) \left(\frac{p}{q}\right)^9

 

 
Practice 6

Use the exponent laws to simplify. If possible, evaluate.

a) 70 b) 3(5)0 c) (–4)1

Homework

  1. Fill in the blanks.

    a) The power 27 has a base of                  and an exponent of                .
    b) The power (–8)2 has a base of                  and an exponent of                .
    c) The power \left(\frac{1}{3}\right)^6 has a base of                  and an exponent of                .
    d) The expression 15p4 has a coefficient of                .
    e) The expression –2m5 has a coefficient of                .

  2. Evaluate.

    a) {4}^{3} b) {7}^{1} c) {\left(\dfrac{5}{6}\right)}^{2}
    d) -{5}^{4} e) {\left(-5\right)}^{4} f) {\left(0.63\right)}^{2}

  3. Use the Product Law to simplify.

    a) {x}^{4} \bullet {x}^{2}

     

     

    		 b) {q}^{27} \bullet {q}^{15}

     

     

    		 c) {3}^{10} \bullet {3}^{6}

     

     

    		 d) 5 \bullet {5}^{4}

     

     
    e) y \bullet {y}^{3}

     

     

    		 f) {z}^{25} \bullet {z}^{8}

     

     

    		 g) y \bullet {y}^{3} \bullet {y}^{5}

     

     

    		 h) {c}^{5} \bullet {c}^{11} \bullet {c}^{2}

     

     

  4. Use the Quotient Law to simplify.

    a) \frac{{y}^{20}}{{y}^{10}}

     

     

    		 b) \frac{{7}^{16}}{{7}^{2}}

     

     

    		 c) \frac{{u}^{24}}{{u}^{3}}

     

     

    		 d) \frac{{9}^{15}}{{9}^{5}}

     

     
    e) \frac{{t}^{40}}{{t}^{10}}

     

     

    		 f) \frac{{8}^{5}}{{8}^{3}}

     

     

    		 g) \frac{{z}^{5}}{{z}^{3}}

     

     

    		 h) \frac{{10}^{3}}{{10}^{2}}

     

     

  5. Use the Power of Power Law to simplify.

    a) {\left({m}^{4}\right)}^{2}

     

    		 b) {\left({10}^{3}\right)}^{6}

     

    		 c) {\left({b}^{2}\right)}^{7}

     

    		 d) {\left({3}^{8}\right)}^{2}

     
    e) {\left({y}^{3}\right)}^{5}

     

    		 f) {\left({5}^{1}\right)}^{7}

     

    		 g) {\left({x}^{9}\right)}^{3}

     

    		 h) {\left({7}^{4}\right)}^{2}

     

  6. Use the Power of Product Law to simplify.

    a) {\left(6a\right)}^{2}

     

     

     

    		 b) {\left(3xy\right)}^{2}

     

     

     

    		 c) {\left(-4m\right)}^{3}

     

     

     

    		 d) {\left(5ab\right)}^{3}

     

     

     
    e) {\left(5x\right)}^{2}

     

     

     

    		 f) {\left(4ab\right)}^{2}

     

     

     

    		 g) {\left(-7n\right)}^{3}

     

     

     

    		 h) {\left(3xyz\right)}^{4}

     

     

     

  7. Use the Power of Quotient Law to simplify.

    a) \frac{{\left({p}^{3}\right)}^{4}}{{p}^{5}}

     

     

     

    		 b) \frac{{\left({x}^{4}\right)}^{5}}{{x}^{15}}

     

     

     

    		 c) \frac{{v}^{20}}{{\left({v}^{4}\right)}^{5}}

     

     

     

    		 d) \frac{{n}^{24}}{{\left({n}^{2}\right)}^{4}}

     

     

     
    e) {\left(\frac{{q}^{8}}{{q}^{2}}\right)}^{3}

     

     

     

     

    		 f) {\left(\frac{{m}^{7}}{{m}^{4}}\right)}^{4}

     

     

     

     

    		 g) {\left(\frac{a}{{b}^{6}}\right)}^{3}

     

     

     

     

    		 h) {\left(\frac{{y}^{4}}{{z}^{10}}\right)}^{5}

     

     

     

     
  8. True or False:  (x4)(x2) = (x4)2
    Explain your answer.

  9. Use the Product Law to explain why x \bullet x=x^2.

  10. Jorge thinks {\left(\dfrac{1}{2}\right)}^{2} equals 1. What is wrong with his reasoning?
  11. Explain why -{5}^{3}={\left(-5\right)}^{3} but -{5}^{4}\ne {\left(-5\right)}^{4}.

  12. Simplify.

    a) {13}^{0}

     

    		 b) {k}^{0}

     

    		 c) -{15}^{0}

     

    		 d) -\left({15}^{0}\right)

     

    		 e) {\left(6y\right)}^{0}

     
    f) 6{y}^{0}

     

     

    		 g) 7{y}^{0}{\left(17y\right)}^{0}

     

     

    		 h) {\left(-93{c}^{7}{d}^{15}\right)}^{0}

     

     
    i) 15{r}^{0}-22{s}^{0}

     

     

     

    		 j) {\left(15r\right)}^{0}-{\left(22s\right)}^{0}

     

     

     

  13. Juanita thinks the quotient \frac{{a}^{24}}{{a}^{6}} simplifies to {a}^{4}. What is wrong with her reasoning?

Answers

1.

a) 2, 7 b) –8, 2 c) \frac{1}{3}, 6 d) 15 e) –2

2.

a) 64 b) 7 c) \dfrac{25}{36} d) -625 e) 625 f) 0.3969

3.

a) {x}^{6} b) {q}^{42} c) {3}^{16} d) {5}^{5}
e) {y}^{4} f) {z}^{33} g) {y}^{9} h) {c}^{18}

4.

a) {y}^{10} b) {7}^{14} c) {u}^{21} d) {9}^{10}
e) {t}^{30} f) {8}^{2} g) {z}^{2} h) {10}^{1}

5.

a) {m}^{8} b) {10}^{18} c) {b}^{14} d) {3}^{16}
e) {y}^{15} f) {5}^{7} g) {x}^{27} h) {7}^{8}

6.

a) 36{a}^{2} b) 9{x}^{2}{y}^{2} c) -64{m}^{3} d) 125{a}^{3}{b}^{3}
e) 25{x}^{2} f) 16{a}^{2}{b}^{2} g) -343{n}^{3} h) 81{x}^{4}{y}^{4}{z}^{4}

7.

a) {p}^{7} b) {x}^{5} c) 1 d) {n}^{16}
e) {q}^{18} f) {m}^{12} g) \frac{{a}^{3}}{{b}^{18}} h) \frac{{y}^{20}}{{z}^{50}}

8. False (x4)(x2) = x4+2 = x6  (x4)2 = x4×2 = x8

9. You can use the Product Law and add the exponents together to simplify. 1 + 1 = 2

10. The exponent of 2 needs to be applied to the numerator and the denominator. 12 = 1 and 22 = 4, so the simplified fraction would be \frac{1}{4}.

11. If the exponent is odd (for example, 3), both answers will be –125. If the exponent is even (for example, 4), one answer will be –625 and one answer will be 625.

12.

a) 1 b) 1 c) -1 d) -1 e) 1
f) 6 g) 7 h) 1 i) -7 j) 0

13. The exponents need to be subtracted to simplify. 24 – 6 = 18

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Berg, T. (2020). Intermediate algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY-NC-SA 4.0

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

Wang, M. (2018). Key concepts of intermediate level math. BCcampus. https://collection.bccampus.ca/textbooks/key-concepts-of-intermediate-level-math-bccampus-204/, licensed under CC BY 4.0

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