Book cover for Elementary Algebra

Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

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32,192 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces square roots and radicals, emphasizing simplification, estimation, and approximation techniques. Students learn to identify perfect squares, simplify expressions including those with variables, and apply the properties of square roots to real-world problems. Additionally, caution is advised when handling negative numbers and when using radicals in expressions involving sums or differences.

Learning Objectives

1

Simplify expressions with square roots, including those with numerical and variable radicands.

2

Estimate and approximate square roots for non-perfect square numbers.

3

Apply the product and quotient properties of square roots to remove perfect square factors.

4

Distinguish between the principal (positive) square root and the negative square root.

5

Utilize square root concepts in real-world applications such as calculating fall times and determining side lengths in geometric problems.

Key Concepts

CONCEPT

DEFINITION

Square Root

If n² = m, then n is a square root of m. The principal square root (denoted by √m) is the non-negative value whose square is m.

Perfect Square

A number that is the square of an integer. For example, 36 is a perfect square because 6² = 36.

Radicand

The number or expression inside the radical sign.

Product Property of Square Roots

For non-negative real numbers a and b, √(a·b) = √a · √b. This property is used to simplify square roots by factoring out perfect squares.

Approximation

An estimated numerical value of a square root when the radicand is not a perfect square, typically rounded to a specified number of decimal places.

Example Problems

Example 1

In the following exercises, simplify. $$ \sqrt{36} $$

Example 2

In the following exercises, simplify. $$ \sqrt{4} $$

Example 3

In the following exercises, simplify. $$ \sqrt{64} $$

Example 4

In the following exercises, simplify. $$ \sqrt{169} $$

Example 5

In the following exercises, simplify. $$ \sqrt{9} $$
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Step-by-Step Explanations

QUESTION

How do you simplify √36?

STEP-BY-STEP ANSWER:

Step 1: Recognize that 36 is a perfect square since 6 × 6 = 36.
Step 2: Therefore, √36 = 6.
Final Answer: 6

Simplifying √36

QUESTION

Estimate √60 by identifying the perfect squares closest to 60.

STEP-BY-STEP ANSWER:

Step 1: Identify the two perfect squares around 60: 49 (with √49 = 7) and 64 (with √64 = 8).
Step 2: Since 60 is between 49 and 64, √60 is between 7 and 8.
Step 3: A closer estimate, if needed, might be approximately 7.75 (using a calculator approximation).
Final Answer: Approximately 7.75 (or simply between 7 and 8)

Estimating √60

QUESTION

Simplify the expression √(9x²).

STEP-BY-STEP ANSWER:

Step 1: Recognize that 9x² is a perfect square because (3x)² = 9x².
Step 2: Thus, √(9x²) = 3x, assuming x is non-negative.
Final Answer: 3x

Simplifying √(9x²)

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Common Mistakes

  • Assuming that the square root of a sum equals the sum of the square roots (e.g., ?(a + b) ? ?a + ?b).
  • Forgetting to apply the restriction that the variable under a square root is non-negative when dealing with the principal square root.
  • Not correctly identifying and factoring out perfect square factors in order to simplify the radicand.
  • Misplacing the negative sign when interpreting square roots of perfect square numbers (confusing -?m with ?m).