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nth Roots

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nth Roots

Study Guide

Key Definition

The nth root of a number $b$ can be expressed as $\sqrt[n]{b}$ and represents a number $x$ such that $x^n = b$.

Important Notes

If $n$ is even, \sqrt[n]{b} exists for $b \geq 0$.
If $n$ is odd, $\sqrt[n]{b}$ exists for all $b$.
$\sqrt[n]{b}$ may not always be a rational number.
Fractional exponent form: $b^{\frac{1}{n}}$.
Example: $\sqrt[3]{64} = 4$ because $4^3 = 64$.

Mathematical Notation

\sqrt[n]{b} means the nth root of $b$

$+$ represents addition

Remember to use proper notation when solving problems

Why It Works

The nth root $\sqrt[n]{b}$ is defined such that raising it to the power of $n$ returns $b$, i.e., $(\sqrt[n]{b})^n = b$.

Remember

The nth root can be expressed as a fractional exponent: $b^{\frac{1}{n}}$.

Quick Reference

Fractional Exponent:$b^{\frac{1}{n}}$

Understanding nth Roots

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The nth root of a number $b$ is the number $x$ such that $x^n = b$. For example, \sqrt[2]{9} = 3.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the value of \sqrt[4]{81}?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A cube has a volume of $64$ cubic units. What is the length of one side of the cube?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Calculate the value of $\sqrt[5]{-32}$.

Challenge Quiz

Single Choice Quiz

Advanced

What is the value of $\left(\sqrt[3]{27}\right)^2$?

Recap

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