Introduction to Square Roots | Mathematics for Digital SAT PDF Download

Introduction

Square roots are an essential concept in algebra and a crucial topic to master for the SAT exam. Understanding square roots and their properties will help you solve a variety of problems, including simplifying expressions, solving equations, and working with radicals. This document will provide a comprehensive overview of square roots, including detailed explanations of concepts, examples, and practice questions with solutions.

1. Definition of Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol √, followed by the number. For example, the square root of 9 is 3 (√9 = 3), since 3 x 3 = 9.

2. Properties of Square Roots

(a) The square root of a product is equal to the product of the square roots:
√(a x b) = √a x √b

Example: √(2 x 8) = √(16) = 4
This is the same as: √2 x √8 = 2 x 2 = 4

(b) The square root of a quotient is equal to the quotient of the square roots:
√(a / b) = √a / √b

Example: √(9 / 4) = √(2.25) = 1.5
This is the same as: √9 / √4 = 3 / 2 = 1.5

(c) The square of a square root eliminates the square root:

(√a)² = a

Example: (√5)² = 5

3. Simplifying Square Roots

To simplify a square root, find the prime factorization of the number inside the square root and identify any pairs of the same number. Each pair can be simplified to the number itself.

Example: Simplify √72
Prime factorization of 72 : 2 x 2 x 2 x 3 x 3
Identify pairs: (2 x 2) x 2 x (3 x 3)
Simplify: √72 = √(4 x 2 x 9) = 2 x √2 x 3 = 6√2

4. Solving Equations with Square Roots

To solve an equation containing a square root, isolate the square root term and then square both sides of the equation. This will eliminate the square root and create a new equation to solve.

Example: Solve √(x + 2) = 3
Step 1: Isolate the square root: √(x + 2) = 3
Step 2: Square both sides: (√(x + 2))² = 3²
Step 3: Solve the new equation: x + 2 = 9
Step 4: Solve for x: x = 7

5. Perfect Squares

Perfect squares are numbers whose square roots are integers. In other words, a perfect square is the product of an integer multiplied by itself. For example, 1, 4, 9, 16, and 25 are perfect squares because they have integer square roots (1, 2, 3, 4, and 5, respectively).

Examples: Square Roots:
(a) √4 = 2, because 2 × 2 = 4
(b) √81 = 9, because 9 × 9 = 81
(c) √121 = 11, because 11 × 11 = 121

Practice Questions

Q.1. Simplify √50.

√50 = √(2 * 5²) = 5√2

Q.2. Solve the equation: √(2x - 4) = 4

√(2x - 4) = 4 → (2x - 4) = 16 → 2x = 20 → x = 10

Q.3. If √(x² - 4) = 5, what is the value of x?

x² - 4 = 25 → x² = 29 → x = ±√29

Q.4. Simplify the expression: 2√18 + 3√8

2√18 + 3√8 = 2(3√2) + 3(2√2) = 6√2 + 6√2 = 12√2

Q.5. Solve the equation: 2√(x + 3) - 4 = 6

2√(x + 3) - 4 = 6 → 2√(x + 3) = 10 → √(x + 3) = 5 → x + 3 = 25 → x = 22

Q.6. Is 50 a perfect square? If not, find the closest perfect squares and their square roots.

No, 50 is not a perfect square. The closest perfect squares are 49 and 64, and their square roots are 7 and 8, respectively.