Solved Examples: Understanding Nth Root Notation and Terminology

📚 Quick Study Guide

• ➗ The nth root of a number $x$ is a value that, when raised to the power of $n$, equals $x$.
• 📝 The notation for the nth root is $\sqrt[n]{x}$, where $n$ is the index (or root) and $x$ is the radicand.
• 🔢 When $n = 2$, it's called the square root, and we usually omit the index: $\sqrt{x}$.
• ✨ When $n = 3$, it's called the cube root: $\sqrt[3]{x}$.
• 💡 If $n$ is even, $x$ must be non-negative for the nth root to be a real number.
• ✅ If $n$ is odd, $x$ can be any real number.
• ➕ Examples: $\sqrt[4]{16} = 2$ because $2^4 = 16$, and $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$.

🧪 Practice Quiz

1. What is the value of $\sqrt[3]{27}$?
   1. 1
   2. 3
   3. 9
   4. 81


2. What is the index in the expression $\sqrt[5]{32}$?
   1. 2
   2. 32
   3. 5
   4. 5/2


3. What is the radicand in the expression $\sqrt[4]{81}$?
   1. 4
   2. 81
   3. 3
   4. 20.25


4. What is the value of $\sqrt[2]{144}$?
   1. 6
   2. 12
   3. 72
   4. 144


5. Which of the following expressions is equivalent to $5^{\frac{1}{2}}$?
   1. $\sqrt[5]{2}$
   2. $\sqrt{5}$
   3. $\sqrt[2]{5}$
   4. $\frac{5}{2}$


6. What is the value of $\sqrt[3]{-64}$?
   1. 4
   2. -4
   3. 8
   4. -8


7. What is the value of $\sqrt[5]{0}$?
   1. 1
   2. 5
   3. 0
   4. Undefined