Lesson 4.5

Intro to Nth Roots

Square roots are just the beginning. Nth roots extend the idea: asks "what number, raised to the th power, gives ?"

Introduction

You already know because . A cube root works the same way: because . This lesson generalizes to any index .

Past Knowledge

Square roots from earlier algebra, exponent rules.

Today's Goal

Evaluate nth roots, understand even vs. odd index behavior, and use radical notation.

Future Success

Nth roots lead to simplifying radicals (4.6) and rational exponents (Chapter 14).

Key Concepts

Notation

n = index (the small number)

a = radicand (under the radical)

√ = radical symbol

Even vs. Odd Index

Even index (2, 4, 6…)

Radicand must be ≥ 0. Result is always ≥ 0.

✓ not real ✗

Odd index (3, 5, 7…)

Radicand can be any real number. Result matches sign.

✓ ✓

Worked Examples

Example 1: Perfect Roots

Basic

Evaluate each root.

because

Example 2: Negative Radicands

Intermediate

Evaluate if possible.

Odd index → OK

Even index, neg. radicand → ✗

Odd index → OK

Example 3: Variable Radicands

Advanced

Simplify each expression.

Even index → absolute value ensures non-negative result

Odd index → no absolute value needed

Even index → absolute value

Common Pitfalls

Forgetting Absolute Value

, NOT just . Without the absolute value, you could get a negative answer for .

Even Root of a Negative

is NOT . No real number raised to the 4th power gives a negative result.

Real-Life Applications

The cube root appears in volume calculations: if a cube has volume 64 cm³, each side is cm. Fourth roots appear in physics formulas for radiation and signal decay.