Evaluate Roots

Grade 8 · algebra · 76 practice problems · read aloud

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Evaluating Roots 🔍

Finding a root of a number is the inverse (opposite) operation of raising a number to a power. If you know that 5² = 25, then the square root of 25 is 5! We use this to solve equations and find side lengths in geometry.

Step-by-Step Guide

  1. Identify the type of root. Is it a square root (√), cube root (∛), or higher?
  2. Find the "perfect" version. Ask yourself: "What number, multiplied by itself, gives me the number under the root?"
  3. Evaluate. Write down the answer. For example, √36 = 6 because 6 × 6 = 36.
  4. Check your work! Multiply your answer by itself to see if you get the original number.

Worked Examples

Example 1: Evaluate √81

Step 1: This is a square root.
Step 2: Ask "What number times itself is 81?"
Step 3: 9 × 9 = 81, so √81 = 9.

Example 2: Evaluate ∛64

Step 1: This is a cube root.
Step 2: Ask "What number multiplied by itself three times is 64?"
Step 3: 4 × 4 × 4 = 64, so ∛64 = 4.

Common Mistakes to Avoid ⚠️

Mistake: Thinking that √(a + b) is the same as √a + √b.

Correction: Roots apply to the entire value underneath. √(9 + 16) = √25 = 5, NOT √9 + √16 (which is 3 + 4 = 7).

Mistake: Forgetting that a square root has both a positive and negative solution.

Correction: While the principal (main) square root is positive, the equation x² = 9 has two solutions: x = 3 and x = -3.

Tips & Tricks

Perfect Square Pairs: Memorize the first 12 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. This makes evaluation instant!

Prime Factorization: For trickier roots, break the number down into its prime factors. For √196, factor 196 into 2×2×7×7. Pair the numbers: (2×7) and (2×7), so √196 = 14.

How to Practice

  • Flashcards: Create flashcards with perfect squares and cubes on one side and their roots on the other.
  • Online Quizzes: Find interactive games that test your speed and accuracy with roots.
  • Real-World Problems: If a square garden has an area of 49 m², how long is one side? (Answer: √49 = 7 meters).

Practice problems

6 of the 76, worked through step by step — try them before opening the answer.

1 ∛(64) + √(81) = ?

Hint: First find the cube root of the first number, then the square root of the second number, and finally add them together.

Show the answer

Answer: 13

  1. Understand the problem We need to compute ∛(64) + √(81). ∛(64) means the cube root of 64. √(81) means the square root of 81.
  2. Calculate the cube root of 64 We ask: what number multiplied by itself three times equals 64? 4 × 4 × 4 = 16 × 4 = 64. So ∛(64) = 4.
  3. Calculate the square root of 81 We ask: what number multiplied by itself equals 81? 9 × 9 = 81. So √(81) = 9.
  4. Add the results 4 + 9 = 13.
  5. Final answer ∛(64) + √(81) = 13.

Let's solve step by step.

2 √(64) + ∛(125) = ?

Hint: First determine the principal square root and cube root of each number separately, then add the results together.

Show the answer

Answer: 13

  1. Understand the problem We need to compute: √(64) + ∛(125)
  2. Calculate the square root of 64 The square root of 64 is the number that, when multiplied by itself, gives 64. Since 8 × 8 = 64, √(64) = 8.
  3. Calculate the cube root of 125 The cube root of 125 is the number that, when multiplied by itself three times, gives 125. Since 5 × 5 × 5 = 25 × 5 = 125, ∛(125) = 5.
  4. Add the results √(64) + ∛(125) = 8 + 5 = 13.
  5. Final answer

Let's solve step by step. The correct answer is 13.

3 ∛(125) + √(64) = ?

Hint: First find the cube root of the first number, then the square root of the second number, and finally add the results together.

Show the answer

Answer: 13

  1. Understand the problem. We need to calculate the cube root of 125 and the square root of 64, then add them together.
  2. Calculate the cube root of 125. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. We ask: what number multiplied by itself three times equals 125? 5 * 5 = 25, and 25 * 5 = 125. So, cube root of 125 = 5.
  3. Calculate the square root of 64. The square root of a number is a value that, when multiplied by itself, gives the original number. We ask: what number multiplied by itself equals 64? 8 * 8 = 64. So, square root of 64 = 8.
  4. Add the two results. 5 + 8 = 13.
  5. Final answer. The result is 13.

4 ∛(125) + √(49) = ?

Hint: First identify the cube root of a perfect cube, then find the square root of a perfect square, and finally add the results together.

Show the answer

Answer: 12

  1. Understand the problem. We need to evaluate the cube root of 125 and the square root of 49, then add them together.
  2. Evaluate the cube root of 125. The cube root of a number is the value that, when multiplied by itself three times, gives the original number. We ask: what number cubed equals 125? 5 × 5 × 5 = 25 × 5 = 125. So, ∛(125) = 5.
  3. Evaluate the square root of 49. The square root of a number is the value that, when multiplied by itself, gives the original number. We ask: what number squared equals 49? 7 × 7 = 49. So, √(49) = 7.
  4. Add the two results. 5 + 7 = 12.
  5. State the final answer. The result is 12.

5 ∛(125) + √(81) = ?

Hint: First find the cube root of the first number, then find the square root of the second number, and finally add the results together.

Show the answer

Answer: 14

  1. Understand the problem We need to compute: cube root of 125 plus square root of 81.
  2. Compute cube root of 125 Cube root means: what number multiplied by itself 3 times gives 125? 5 × 5 = 25 25 × 5 = 125 So, cube root of 125 = 5.
  3. Compute square root of 81 Square root means: what number multiplied by itself gives 81? 9 × 9 = 81 So, square root of 81 = 9.
  4. Add the results 5 + 9 = 14.
  5. Final answer The result is 14.

Let's solve step by step.

6 ∛(216) + √(144) = ?

Hint: First find the cube root of the first number, then the square root of the second number, and finally add them together.

Show the answer

Answer: 18

  1. Find the cube root of 216. Since 6 × 6 × 6 = 216, ∛(216) = 6
  2. Find the square root of 144. Since 12 × 12 = 144, √(144) = 12
  3. Add the results: 6 + 12 = 18

The answer is 18.

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