Square Roots

Grade 12 Math - Square Roots

Lesson Objectives

Understand the concept of square roots and perfect squares.
Calculate square roots of perfect and non-perfect squares.
Simplify square roots using prime factorization.
Apply square roots in solving real-world and algebraic problems.

Lesson Introduction

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because $5 \times 5 = 25$. In this lesson, we will explore both perfect and non-perfect square roots and how to simplify or estimate them using different methods.

Core Lesson Content

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. Examples include:

$1^2 = 1,\quad 2^2 = 4,\quad 3^2 = 9,\quad 4^2 = 16,\quad 5^2 = 25,\quad 10^2 = 100$

Square Roots of Perfect Squares

The square root symbol is $\sqrt{}$ . The square root of 36 is:

$\sqrt{36} = 6$

Simplifying Square Roots Using Prime Factorization

Break the number into its prime factors, and group identical pairs.

$\sqrt{72} = \sqrt{2 \times 2 \times 2 \times 3 \times 3} = \sqrt{(2^2) \times (3^2) \times 2} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2}$

Estimating Square Roots

For non-perfect squares, estimate between two perfect squares. For example:

$\sqrt{50} \approx 7.1\quad \text{because } 7^2 = 49 \text{ and } 8^2 = 64$

Worked Examples

Example 1: Find

\sqrt{49}

\sqrt{49} = 7

Example 2: Simplify

\sqrt{75}

\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}

Example 3: Estimate

\sqrt{90}

9^2 = 81, \quad 10^2 = 100 \Rightarrow \sqrt{90} \approx 9.5

Example 4: Simplify

\sqrt{200}

using prime factorization.

\sqrt{200} = \sqrt{2 \times 2 \times 2 \times 5 \times 5} = 2 \times 5 \sqrt{2} = 10\sqrt{2}

Example 5: Find

\sqrt{0.04}

\sqrt{0.04} = 0.2

Example 6: Express

\sqrt{18}

in its simplest radical form.

\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}

Example 7: Find

\sqrt{225}

\sqrt{225} = 15

Example 8: Estimate

\sqrt{115}

10^2 = 100, 11^2 = 121 \Rightarrow \sqrt{115} \approx 10.7

Example 9: Find the square root of

1.44

\sqrt{1.44} = 1.2

Example 10: Simplify

\sqrt{98}

\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}

Exercises

Find $\sqrt{121}$ .
Simplify $\sqrt{32}$ .
[JAMB] Estimate $\sqrt{80}$ . (Past Question)
Use prime factorization to simplify $\sqrt{180}$ .
[WAEC] Find the square root of $0.09$ . (Past Question)
Simplify $\sqrt{50} + \sqrt{18}$ .
[NECO] Estimate $\sqrt{65}$ . (Past Question)
If $x^2 = 81$ , find the value of $x$ .
Simplify $\sqrt{147}$ .
[WAEC] Estimate $\sqrt{135}$ to one decimal place. (Past Question)

Conclusion/Recap

In this lesson, we explored how to calculate square roots, recognize perfect squares, and simplify complex square roots using factorization and estimation. Mastery of square roots is crucial in solving quadratic equations, geometry problems, and many real-world applications. .