N-th Roots and Rational Exponents: Review and Mastery

Review of Algebra

N-th Roots and Rational Exponents

This section provides a comprehensive review of n-th roots and rational exponents, foundational concepts in algebra that are essential for simplifying expressions and solving equations. Mastery of these topics is crucial for success in College Algebra and subsequent mathematics courses.

Objectives

• Work with n-th roots

• Simplify radicals

• Rationalize denominators and numerators

• Simplify expressions with rational exponents

Definition: N-th Root

The n-th root of a number a is a value that, when raised to the n-th power, equals a. The principal n-th root is denoted as $\sqrt[n]{a}$.

• If n is even, $\sqrt[n]{a}$ is defined only for $a \geq 0$.

• If n is odd, $\sqrt[n]{a}$ is defined for all real numbers a.

Example 1

Find the cube root of 27.

Solution: $\sqrt[3]{27} = 3$ because $3^3 = 27$.

Simplifying Radicals

To simplify a radical, factor the radicand (the number under the radical sign) into its prime factors and extract powers that match the index.

• Example: $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

Rationalizing Denominators and Numerators

Rationalizing means rewriting a fraction so that there are no radicals in the denominator or numerator. This is often done by multiplying by a form of 1 that eliminates the radical.

• Example: $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Definition: Rational Exponents

A rational exponent expresses roots and powers in a single notation. For any positive real number a and integers m and n ($n > 0$):

• $a^{1/n} = \sqrt[n]{a}$

• $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$

Example 2

Express $8^{2/3}$ as a radical and simplify.

Solution: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$

Simplifying Expressions with Rational Exponents

To simplify expressions with rational exponents, rewrite them using radical notation and apply the properties of exponents and radicals.

• Example: $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$

Properties of Exponents and Radicals

• $a^{m} \cdot a^{n} = a^{m+n}$

• $(a^{m})^{n} = a^{mn}$

• $(ab)^{n} = a^{n}b^{n}$

• $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$

Table: Common Roots and Their Equivalent Rational Exponents

Radical Form	Rational Exponent Form	Example
$\sqrt{a}$	$a^{1/2}$	$\sqrt{9} = 9^{1/2} = 3$
$\sqrt[3]{a}$	$a^{1/3}$	$\sqrt[3]{8} = 8^{1/3} = 2$
$\sqrt[n]{a^m}$	$a^{m/n}$	$\sqrt[4]{16^3} = 16^{3/4} = 8$

Additional info:

• Understanding the relationship between radicals and rational exponents is essential for manipulating algebraic expressions and solving equations involving roots and powers.

• These skills are foundational for topics such as polynomial and rational functions, exponential and logarithmic functions, and more advanced mathematics.