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Exponent Rules and Properties

Introduction to Exponents

Exponents are a fundamental concept in algebra, representing repeated multiplication of a base number. Understanding the properties and rules of exponents is essential for simplifying expressions and solving equations in College Algebra.

• Exponent notation: $b^n$ means multiplying the base $b$ by itself $n$ times.

• Base: The number being multiplied.

• Exponent: The number of times the base is used as a factor.

• Example: $3^4 = 3 \times 3 \times 3 \times 3 = 81$

Basic Properties of Exponents

Several key rules govern how exponents behave in algebraic expressions. These rules allow us to simplify and manipulate expressions efficiently.

• Product Rule: When multiplying like bases, add the exponents. $b^m \cdot b^n = b^{m+n}$

• Quotient Rule: When dividing like bases, subtract the exponents. $\dfrac{b^m}{b^n} = b^{m-n}$, $b \neq 0$

• Power Rule: When raising a power to another power, multiply the exponents. $(b^m)^n = b^{mn}$

• Power of a Product Rule: Distribute the exponent to each factor inside the parentheses. $(ab)^n = a^n b^n$

• Power of a Quotient Rule: Distribute the exponent to both numerator and denominator. $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$, $b \neq 0$

Zero and Negative Exponents

Exponents can be zero or negative, and these cases have special rules.

• Zero Exponent Rule: Any nonzero base raised to the zero power is 1. $b^0 = 1$, $b \neq 0$

• Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the positive exponent. $b^{-n} = \dfrac{1}{b^n}$, $b \neq 0$

• Example: $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$

Rational (Fractional) Exponents

Rational exponents represent roots. The denominator of the exponent indicates the root, and the numerator indicates the power.

• Rational Exponent Rule: $b^{\frac{m}{n}} = \sqrt[n]{b^m} = (\sqrt[n]{b})^m$

• Example: $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$

• Additional info: The rule does not work if $n$ is even and $b < 0$ (since even roots of negative numbers are not real).

Simplifying Exponential Expressions

To simplify expressions with exponents, apply the rules above systematically.

• Example 1: $\dfrac{a^7}{a^3} = a^{7-3} = a^4$

• Example 2: $(2^3)^4 = 2^{3 \times 4} = 2^{12}$

• Example 3: $\left(\dfrac{x^2 y^3}{z^4}\right)^2 = \dfrac{x^{2 \times 2} y^{3 \times 2}}{z^{4 \times 2}} = \dfrac{x^4 y^6}{z^8}$

Solving Exponential Equations

Equations involving exponents can often be solved by expressing both sides with the same base and applying exponent rules.

• Example: Solve $7^x = 49$. Since $49 = 7^2$, set $7^x = 7^2$ so $x = 2$.

• Example: Solve $2^{x+2} = 2^5$. Set exponents equal: $x+2 = 5$ so $x = 3$.

Summary Table: Exponent Rules

Rule Name	Formula	Conditions
Product Rule	$b^m \cdot b^n = b^{m+n}$	Any $b$
Quotient Rule	$\dfrac{b^m}{b^n} = b^{m-n}$	$b \neq 0$
Power Rule	$(b^m)^n = b^{mn}$	Any $b$
Power of a Product	$(ab)^n = a^n b^n$	Any $a, b$
Power of a Quotient	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$	$b \neq 0$
Zero Exponent	$b^0 = 1$	$b \neq 0$
Negative Exponent	$b^{-n} = \dfrac{1}{b^n}$	$b \neq 0$
Rational Exponent	$b^{\frac{m}{n}} = \sqrt[n]{b^m}$	$n$ even, $b \geq 0$

Practice Problems

• Simplify: $(a^2 b^{-3})^4$

• Solve: $x^{1/2} = 5$

• Simplify: $\dfrac{2x^3 y^{-2}}{4x^{-1} y^4}$

Conclusion

Mastering exponent rules is crucial for success in College Algebra. These properties are foundational for manipulating algebraic expressions, solving equations, and understanding more advanced mathematical concepts.