Introduction

Square Roots and Cube Roots is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with square roots and cube roots.

What Is Square Roots and Cube Roots?

Square Roots and Cube Roots means measuring how much flat space a figure covers by using equal-sized square units.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Square Roots and Cube Roots

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Use square units that cover the figure without gaps or overlaps.
  • Count rows and columns when the unit squares are arranged in an array.
  • Connect repeated addition to multiplication when finding area.
  • Break complex figures into smaller rectangles when that makes the work clearer.

Visual Models

Visual Model 1

Question: A square has an area of \(361\) square cm. What is its perimeter?

Visual Model 1

  • A. \(38\) cm
  • B. \(57\) cm
  • C. \(76\) cm
  • D. \(152\) cm

Why it works: Area \(= 361\), so \(s = \sqrt{361} = 19\) cm. Perimeter \(= 4s = 4 \times 19 = 76\) cm.

Answer: \(76\) cm

Visual Model 2

Question: If a cube has a side length of \(5\) cm, what is its volume?

Visual Model 2

  • A. \(15\) cm\(^3\)
  • B. \(25\) cm\(^3\)
  • C. \(75\) cm\(^3\)
  • D. \(125\) cm\(^3\)

Why it works: Volume of a cube \(= s^3 = 5^3 = 125\) cm\(^3\).

Answer: \(125\) cm\(^3\)

Worked Examples

Example 1

Question: Between which two consecutive integers does \(\sqrt[3]{50}\) lie?

Example 1

  • A. \(2\) and \(3\)
  • B. \(5\) and \(6\)
  • C. \(4\) and \(5\)
  • D. \(3\) and \(4\)
  1. \(3^3 = 27\) and \(4^3 = 64\).
  2. Since \(27 < 50 < 64\), we have \(3 < \sqrt[3]{50} < 4\).

Answer: \(3\) and \(4\)

Example 2

Question: Which is a reasonable approximation for \(\sqrt{200}\)?

Example 2

  • A. \(12\)
  • B. \(18\)
  • C. \(16\)
  • D. \(14\)
  1. \(\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \approx 10 \times 1.41 = 14.1\).
  2. So \(14\) is the closest integer.

Answer: \(14\)

Example 3

Question: If the volume of a cube is \(343\) cm\(^3\), what is the surface area of one face?

Example 3

  • A. \(7\) cm\(^2\)
  • B. \(294\) cm\(^2\)
  • C. \(98\) cm\(^2\)
  • D. \(49\) cm\(^2\)
  1. Volume \(= s^3 = 343\), so \(s = \sqrt[3]{343} = 7\) cm.
  2. One face is a square with side \(7\) cm, so area \(= 7^2 = 49\) cm\(^2\).

Answer: \(49\) cm\(^2\)

Real-World Word Problems

Problem 1

Question: A square picture frame has an area of \(169\) square inches. What is the length of one side of the frame?

  • A. \(12\) inches
  • B. \(15\) inches
  • C. \(14\) inches
  • D. \(13\) inches

Why it works: If the area is \(169\) square inches, then \(s^2 = 169\), so \(s = \sqrt{169} = 13\) inches.

Answer: \(13\) inches

Problem 2

Question: A rectangular garden has an area of \(200\) square meters. If it is a square, what is the approximate side length?

  • A. \(13\) meters
  • B. \(16.2\) meters
  • C. \(15\) meters
  • D. \(14.1\) meters

Why it works: \(s = \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \approx 14.14\) meters.

Answer: \(14.1\) meters

Common Mistakes

  • Counting only the outside squares instead of all squares inside the figure.
  • Leaving gaps or overlaps when using unit squares.
  • Multiplying side lengths before checking whether the figure is a rectangle.
  • Forgetting to write square units with an area answer.

Strategy Tips

  • Trace the rectangle or figure before counting.
  • Use rows and columns to organize unit squares.
  • Write an equation after the model makes sense.
  • Check whether the answer needs square units.

Practice Questions

Question 1

What is \(\sqrt[3]{125}\)?

  • A. \(25\)
  • B. \(11\)
  • C. \(15\)
  • D. \(5\)

Question 2

Which of the following is a perfect square?

  • A. \(48\)
  • B. \(99\)
  • C. \(75\)
  • D. \(64\)

Question 3

What is \(\sqrt{400}\)?

  • A. \(18\)
  • B. \(19\)
  • C. \(20\)
  • D. \(21\)

Question 4

Which number is closest to \(\sqrt{50}\)?

  • A. \(6\)
  • B. \(7\)
  • C. \(7.1\)
  • D. \(9\)

Question 5

What is \(\sqrt[3]{216}\)?

  • A. \(5\)
  • B. \(8\)
  • C. \(7\)
  • D. \(6\)

Question 6

If \(x^2 = 144\), what is the positive value of \(x\)?

  • A. \(11\)
  • B. \(14\)
  • C. \(13\)
  • D. \(12\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(5\)

\(5\times5\times5=125\), so \(\sqrt[3]{125}=5\).

Question 2

Answer: \(64 = 8 \times 8 = 8^2\)

\(\sqrt{64} = 8\). The other numbers are not perfect squares.

Question 3

Answer: \(20\)

\(20 \times 20 = 400\), so \(\sqrt{400} = 20\).

Question 4

Answer: \(7.1\)

\(7^2 = 49\) and \(8^2 = 64\), so \(\sqrt{50}\) is a little more than \(7\). Since \(\sqrt{50} \approx 7.07\), it is closest to \(7.1\).

Question 5

Answer: \(6\)

\(6 \times 6 \times 6 = 216\), so \(\sqrt[3]{216} = 6\).

Question 6

Answer: \(12\)

Solve \(x^2 = 144\) by taking the positive square root: \(x = \sqrt{144} = 12\).

Timed practice quizzes

Related Quizzes

After studying Square Roots and Cube Roots, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Square Roots and Cube Roots becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Area means every square unit inside the figure.