C1

Equations & Inequalities

Properties of Exponents

Understanding the properties of exponents is essential for simplifying expressions and solving equations in algebra. These rules allow us to rewrite expressions in simpler or more useful forms.

• Exponent: In an expression of the form $b^n$, $b$ is the base and $n$ is the exponent.

• Exponential Form: Repeated multiplication can be written as $b^n = b \cdot b \cdot \ldots \cdot b$ ($n$ times).

Basic Properties of Exponents

• Product Rule: $b^m \cdot b^n = b^{m+n}$

• Quotient Rule: $\dfrac{b^m}{b^n} = b^{m-n}$, $b \neq 0$

• Power Rule: $(b^m)^n = b^{mn}$

• Power of a Product Rule: $(ab)^n = a^n b^n$

• Power of a Quotient Rule: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$, $b \neq 0$

• Zero Exponent Rule: $b^0 = 1$, $b \neq 0$

• Negative Exponent Rule: $b^{-n} = \dfrac{1}{b^n}$ and $\dfrac{1}{b^{-n}} = b^n$

• Rational Exponent Rule: $b^{1/n} = \sqrt[n]{b}$, $b \geq 0$ for even $n$

• General Rational Exponent: $b^{m/n} = \sqrt[n]{b^m}$

Examples and Applications

• Example 1: Simplify $\dfrac{16x^7}{2x^3}$ $\dfrac{16}{2} \cdot \dfrac{x^7}{x^3} = 8x^{7-3} = 8x^4$

• Example 2: Rewrite $8 \cdot 4^x$ as a single factor with a base of 2 $8 = 2^3$, $4 = 2^2$ so $8 \cdot 4^x = 2^3 \cdot (2^2)^x = 2^3 \cdot 2^{2x} = 2^{2x+3}$

• Example 3: Solve $7^x = 49$ $49 = 7^2$, so $7^x = 7^2 \implies x = 2$

• Example 4: Simplify $\dfrac{a^7 b^3}{a^2 b^5}$ $a^{7-2} b^{3-5} = a^5 b^{-2} = \dfrac{a^5}{b^2}$

• Example 5: Simplify $\left(\dfrac{3^2}{5^3}\right)^4$ $\dfrac{3^{2\cdot4}}{5^{3\cdot4}} = \dfrac{3^8}{5^{12}}$

• Example 6: Solve $2^{x+2} = 32$ $32 = 2^5$, so $2^{x+2} = 2^5 \implies x+2 = 5 \implies x = 3$

• Example 7: Simplify $\left(\dfrac{5x^2}{y^3}\right)^2$ $\dfrac{5^2 x^{2\cdot2}}{y^{3\cdot2}} = \dfrac{25x^4}{y^6}$

Special Exponent Cases

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

• Negative Exponent: A negative exponent indicates a reciprocal. $b^{-n} = \dfrac{1}{b^n}$

• Fractional Exponent: A rational exponent represents a root. $b^{1/n} = \sqrt[n]{b}$

• General Rational Exponent: $b^{m/n} = \sqrt[n]{b^m}$

Exponent Rules Table

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	$b^{1/n} = \sqrt[n]{b}$	$b \geq 0$ for even $n$
General Rational Exponent	$b^{m/n} = \sqrt[n]{b^m}$	$b \geq 0$ for even $n$

Practice Problems

• Simplify: $a^3 b^{-2} = \dfrac{a^3}{b^2}$

• Solve: $x^{1/2} = 4 \implies x = 16$

• Simplify: $\left(\dfrac{x^2}{y^3}\right)^{-2} = \dfrac{y^6}{x^4}$

Summary

• Mastering exponent rules is crucial for simplifying algebraic expressions and solving equations.

• Always write final answers with positive exponents unless otherwise specified.

• Be careful with negative and fractional exponents; rewrite them using roots and reciprocals as needed.