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Chapter 7: Problem 2

Find each square root. \(\sqrt{0.25}\) (b) \(\sqrt{0.81}\) (c) \(\sqrt{1.44}\)

Short Answer

Expert verified
Square roots are: 0.5, 0.9, 1.2.

Step by step solution

01

Understand the Problem

We are tasked with finding the square roots of the given numbers. A square root of a number is a value that, when multiplied by itself, gives the original number.
02

Calculate \sqrt{0.25}

Identify the number whose square is 0.25. Since \(0.5 \times 0.5 = 0.25\), we have \sqrt{0.25} = 0.5.
03

Calculate \sqrt{0.81}

Identify the number whose square is 0.81. Since \(0.9 \times 0.9 = 0.81\), we have \sqrt{0.81} = 0.9.
04

Calculate \sqrt{1.44}

Identify the number whose square is 1.44. Since \(1.2 \times 1.2 = 1.44\), we have \sqrt{1.44} = 1.2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculating square roots
Calculating square roots involves finding a number which, when multiplied by itself, gives the original number. For example, to find the square root of 0.25, you need to identify a number that, when squared, equals 0.25. In this case, it's 0.5 because \(0.5 \times 0.5 = 0.25\).

Here鈥檚 a brief guide to calculating square roots:
  • Identify the number whose square equals the given number.
  • Use the property of squares: \( a \times a = a^2 \), where \( a \) is the square root.
  • Double-check your results by squaring them again to confirm.
As seen in the example, \( \sqrt{0.25} = 0.5 \) because \( 0.5 \times 0.5 \) gives back 0.25.
Basic arithmetic
Understanding basic arithmetic is crucial for successfully calculating square roots. Basic arithmetic includes operations such as addition, subtraction, multiplication, and division. When dealing with square roots, multiplication is the most extensively used operation.

For example, to verify the square root of 0.81:
  • Identify that \( 0.9 \times 0.9 = 0.81 \), thus \( \sqrt{0.81} = 0.9 \).
  • Practice multiplication of various numbers to become more comfortable with these operations.
Always ensure to break down the problem step-by-step and maintain a consistent practice routine with basic arithmetic to strengthen your understanding.
Square root properties
Square roots have specific properties that help in simplifying and understanding them better. Here are some critical properties of square roots:
  • The square root of 1 is 1, i.e., \( \sqrt{1} = 1 \).
  • The square root of a product is the product of the square roots, i.e., \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).
  • Every positive number has two square roots: one positive and one negative, e.g., both 0.9 and -0.9 are square roots of 0.81.
Utilizing these properties helps in solving square roots more efficiently. For instance, in calculating \( \sqrt{1.44} \), you recognize that \( 1.2 \times 1.2 = 1.44 \), therefore, \( \sqrt{1.44} = 1.2 \), while also keeping in mind that -1.2 is another square root.

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Most popular questions from this chapter

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{27}}{3-\sqrt{3}} $$

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{\frac{2}{3}} $$

Rationalize each denominator. Assume that all radicals represent real numbers and that no denominators are \(0 .\) $$ \frac{5}{\sqrt{m-n}} $$

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[3]{x z} \cdot \sqrt{z} $$

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[5]{x^{3}} \cdot \sqrt[4]{x} $$
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