Learn on PengiBig Ideas Math, Advanced 2Chapter 10: Exponents and Scientific Notation

Section 10.3: Quotient of Powers Property

In this Grade 7 lesson from Big Ideas Math Advanced 2, students learn the Quotient of Powers Property, which states that dividing two powers with the same base requires subtracting the exponents (a^m ÷ a^n = a^(m−n)). Through guided activities and worked examples, students practice simplifying expressions involving quotients of powers, including cases that also apply the Product of Powers Property. The lesson builds fluency with algebraic expressions containing variable and integer bases.

Section 1

Quotient property of exponents

Property

If aa is a real number, a≠0a \neq 0, and mm, nn are whole numbers, then

aman=am−n,m>n and aman=1an−m,n>m\frac{a^m}{a^n} = a^{m-n}, \quad m > n \text{ and } \frac{a^m}{a^n} = \frac{1}{a^{n-m}}, \quad n > m

Examples

  • To simplify x9x4\frac{x^9}{x^4}, you subtract the exponents since 9>49 > 4. The result is x9−4=x5x^{9-4} = x^5.
  • To simplify c5c8\frac{c^5}{c^8}, the larger exponent is in the denominator, so the result is 1c8−5=1c3\frac{1}{c^{8-5}} = \frac{1}{c^3}.

Section 2

Quotients of powers

Property

To divide two powers with the same base, we subtract the smaller exponent from the larger one, and keep the same base.

  1. If the larger exponent occurs in the numerator, put the power in the numerator.
  2. If the larger exponent occurs in the denominator, put the power in the denominator.

In symbols:

  1. aman=am−n\frac{a^m}{a^n} = a^{m-n} if n<mn < m
  2. aman=1an−m\frac{a^m}{a^n} = \frac{1}{a^{n-m}} if n>mn > m

Book overview

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Chapter 10: Exponents and Scientific Notation

  1. Lesson 1

    Section 10.2: Product of Powers Property

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  2. Lesson 2

    Section 10.4: Zero and Negative Exponents

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  3. Lesson 3

    Section 10.7: Operations in Scientific Notation

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Lesson overview

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Section 1

Quotient property of exponents

Property

If aa is a real number, a≠0a \neq 0, and mm, nn are whole numbers, then

aman=am−n,m>n and aman=1an−m,n>m\frac{a^m}{a^n} = a^{m-n}, \quad m > n \text{ and } \frac{a^m}{a^n} = \frac{1}{a^{n-m}}, \quad n > m

Examples

  • To simplify x9x4\frac{x^9}{x^4}, you subtract the exponents since 9>49 > 4. The result is x9−4=x5x^{9-4} = x^5.
  • To simplify c5c8\frac{c^5}{c^8}, the larger exponent is in the denominator, so the result is 1c8−5=1c3\frac{1}{c^{8-5}} = \frac{1}{c^3}.

Section 2

Quotients of powers

Property

To divide two powers with the same base, we subtract the smaller exponent from the larger one, and keep the same base.

  1. If the larger exponent occurs in the numerator, put the power in the numerator.
  2. If the larger exponent occurs in the denominator, put the power in the denominator.

In symbols:

  1. aman=am−n\frac{a^m}{a^n} = a^{m-n} if n<mn < m
  2. aman=1an−m\frac{a^m}{a^n} = \frac{1}{a^{n-m}} if n>mn > m

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 10: Exponents and Scientific Notation

  1. Lesson 1

    Section 10.2: Product of Powers Property

    LearnPractice
  2. Lesson 2

    Section 10.4: Zero and Negative Exponents

    LearnPractice
  3. Lesson 3

    Section 10.7: Operations in Scientific Notation

    LearnPractice