Learn on PengiBig Ideas Math, Algebra 1Chapter 6: Exponential Functions and Sequences

Lesson 1: Properties of Exponents

Property Zero as an Exponent $$a^0 = 1, \quad \text{if } a \neq 0$$.

Section 1

Negative and Zero Exponents

Property

Zero as an Exponent

a0=1,if a≠0a^0 = 1, \quad \text{if } a \neq 0

Negative Exponents

a−n=1anif a≠0a^{-n} = \frac{1}{a^n} \quad \text{if } a \neq 0

A negative power is the reciprocal of the corresponding positive power. A negative exponent does not mean that the power is negative. The laws of exponents apply to negative exponents.

Section 2

Using the Laws of Exponents

Property

To simplify complex expressions, combine the laws of exponents while following the order of operations. Always simplify powers before performing multiplication.

  1. am⋅an=am+na^m \cdot a^n = a^{m+n}
  2. aman=am−n\frac{a^m}{a^n} = a^{m-n} or 1an−m\frac{1}{a^{n-m}}
  3. (am)n=amn(a^m)^n = a^{mn}
  4. (ab)n=anbn(ab)^n = a^nb^n
  5. (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}

Examples

  • Simplify 3a2b(2ab2)33a^2b(2ab^2)^3. First, cube the term in parentheses: 3a2b(8a3b6)3a^2b(8a^3b^6). Then multiply: 24a2+3b1+6=24a5b724a^{2+3}b^{1+6} = 24a^5b^7.
  • Simplify (−y)2(−yz)3(-y)^2(-yz)^3. Simplify each power first: y2(−y3z3)y^2(-y^3z^3). Then multiply: −y2+3z3=−y5z3-y^{2+3}z^3 = -y^5z^3.
  • Simplify (x42)2(3x)2(\frac{x^4}{2})^2(3x)^2. Simplify powers: (x84)(9x2)(\frac{x^8}{4})(9x^2). Then multiply: 9x8+24=9x104\frac{9x^{8+2}}{4} = \frac{9x^{10}}{4}.

Explanation

When expressions have multiple operations, always follow the order of operations (PEMDAS). Simplify any powers first, such as (3x2)3(3x^2)^3, before you multiply that result by other terms in the expression.

Section 3

Multi-Step Exponent Evaluation

Property

When evaluating complex expressions with multiple exponent properties, apply a systematic approach:

  • simplify expressions within parentheses first,
  • apply power of a power/product rules to remove parentheses,
  • use product and quotient rules to combine identical bases, and
  • finally rewrite the expression so it only contains positive exponents.

Examples

  • Evaluate (2x3)2⋅x−1x4\frac{(2x^3)^2 \cdot x^{-1}}{x^4}:

First apply power of a product: 4x6⋅x−1x4\frac{4x^6 \cdot x^{-1}}{x^4}
Then product rule (numerator): 4x5x4\frac{4x^5}{x^4}
Finally quotient rule: 4x1=4x4x^1 = 4x

  • Simplify (3a−2b4)2⋅(ab)−3(3a^{-2}b^4)^2 \cdot (ab)^{-3}:

Apply power of a product to both terms: 9a−4b8⋅a−3b−39a^{-4}b^8 \cdot a^{-3}b^{-3}
Combine like bases (add exponents): 9a−7b59a^{-7}b^5
Rewrite with positive exponents: 9b5a7\frac{9b^5}{a^7}

  • Evaluate (52)−1⋅5450⋅53\frac{(5^2)^{-1} \cdot 5^4}{5^0 \cdot 5^3}:

Simplify powers and zero exponents: 5−2⋅541⋅53=5253=5−1=15\frac{5^{-2} \cdot 5^4}{1 \cdot 5^3} = \frac{5^2}{5^3} = 5^{-1} = \frac{1}{5}

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Chapter 6: Exponential Functions and Sequences

  1. Lesson 1Current

    Lesson 1: Properties of Exponents

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  2. Lesson 2

    Lesson 2: Radicals and Rational Exponents

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  3. Lesson 3

    Lesson 3: Exponential Functions

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  4. Lesson 4

    Lesson 4: Exponential Growth and Decay

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  5. Lesson 5

    Lesson 5: Solving Exponential Equations

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  6. Lesson 6

    Lesson 6: Geometric Sequences

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  7. Lesson 7

    Lesson 7: Recursively Defi ned Sequences

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Section 1

Negative and Zero Exponents

Property

Zero as an Exponent

a0=1,if a≠0a^0 = 1, \quad \text{if } a \neq 0

Negative Exponents

a−n=1anif a≠0a^{-n} = \frac{1}{a^n} \quad \text{if } a \neq 0

A negative power is the reciprocal of the corresponding positive power. A negative exponent does not mean that the power is negative. The laws of exponents apply to negative exponents.

Section 2

Using the Laws of Exponents

Property

To simplify complex expressions, combine the laws of exponents while following the order of operations. Always simplify powers before performing multiplication.

  1. am⋅an=am+na^m \cdot a^n = a^{m+n}
  2. aman=am−n\frac{a^m}{a^n} = a^{m-n} or 1an−m\frac{1}{a^{n-m}}
  3. (am)n=amn(a^m)^n = a^{mn}
  4. (ab)n=anbn(ab)^n = a^nb^n
  5. (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}

Examples

  • Simplify 3a2b(2ab2)33a^2b(2ab^2)^3. First, cube the term in parentheses: 3a2b(8a3b6)3a^2b(8a^3b^6). Then multiply: 24a2+3b1+6=24a5b724a^{2+3}b^{1+6} = 24a^5b^7.
  • Simplify (−y)2(−yz)3(-y)^2(-yz)^3. Simplify each power first: y2(−y3z3)y^2(-y^3z^3). Then multiply: −y2+3z3=−y5z3-y^{2+3}z^3 = -y^5z^3.
  • Simplify (x42)2(3x)2(\frac{x^4}{2})^2(3x)^2. Simplify powers: (x84)(9x2)(\frac{x^8}{4})(9x^2). Then multiply: 9x8+24=9x104\frac{9x^{8+2}}{4} = \frac{9x^{10}}{4}.

Explanation

When expressions have multiple operations, always follow the order of operations (PEMDAS). Simplify any powers first, such as (3x2)3(3x^2)^3, before you multiply that result by other terms in the expression.

Section 3

Multi-Step Exponent Evaluation

Property

When evaluating complex expressions with multiple exponent properties, apply a systematic approach:

  • simplify expressions within parentheses first,
  • apply power of a power/product rules to remove parentheses,
  • use product and quotient rules to combine identical bases, and
  • finally rewrite the expression so it only contains positive exponents.

Examples

  • Evaluate (2x3)2⋅x−1x4\frac{(2x^3)^2 \cdot x^{-1}}{x^4}:

First apply power of a product: 4x6⋅x−1x4\frac{4x^6 \cdot x^{-1}}{x^4}
Then product rule (numerator): 4x5x4\frac{4x^5}{x^4}
Finally quotient rule: 4x1=4x4x^1 = 4x

  • Simplify (3a−2b4)2⋅(ab)−3(3a^{-2}b^4)^2 \cdot (ab)^{-3}:

Apply power of a product to both terms: 9a−4b8⋅a−3b−39a^{-4}b^8 \cdot a^{-3}b^{-3}
Combine like bases (add exponents): 9a−7b59a^{-7}b^5
Rewrite with positive exponents: 9b5a7\frac{9b^5}{a^7}

  • Evaluate (52)−1⋅5450⋅53\frac{(5^2)^{-1} \cdot 5^4}{5^0 \cdot 5^3}:

Simplify powers and zero exponents: 5−2⋅541⋅53=5253=5−1=15\frac{5^{-2} \cdot 5^4}{1 \cdot 5^3} = \frac{5^2}{5^3} = 5^{-1} = \frac{1}{5}

Book overview

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Chapter 6: Exponential Functions and Sequences

  1. Lesson 1Current

    Lesson 1: Properties of Exponents

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  2. Lesson 2

    Lesson 2: Radicals and Rational Exponents

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  3. Lesson 3

    Lesson 3: Exponential Functions

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  4. Lesson 4

    Lesson 4: Exponential Growth and Decay

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  5. Lesson 5

    Lesson 5: Solving Exponential Equations

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  6. Lesson 6

    Lesson 6: Geometric Sequences

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  7. Lesson 7

    Lesson 7: Recursively Defi ned Sequences

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