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Chapter 10: Problem 25

Simplify. $$ \sqrt{0.04} $$

Short Answer

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Step by step solution

01

Understand the square root

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, \(\text{if } \sqrt{x} = y, \text{then } y^2 = x\).
02

Consider the decimal

We need to simplify \(\root 0.04\). Notice that \(\root 0.04\) needs to be converted into a fraction for easier calculation. Notice that \(\root 0.04\) can be written as \(\root \frac{4}{100}\).
03

Simplify the fraction

Recognize that \(\frac{4}{100} = \frac{1}{25}\). Therefore, \(\root 0.04 = \root \frac{1}{25}\).
04

Find the square root of the fraction

The square root of a fraction \(\frac{a}{b}\) is found by taking the square root of both the numerator and the denominator separately. Therefore, \(\root \frac{1}{25} = \root 1 / \root 25 = \frac{1}{5}\).
05

Simplify the expression

Since \(\root 1 = 1\) and \(\root 25 = 5\), \(\frac{1}{5}\) simplifies directly into \(\frac{1}{5} = 0.2\). Therefore, \(\root 0.04 = 0.2\).

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Key Concepts

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

square root
The square root of a number is the value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 multiplied by 5 equals 25. This relationship is written in mathematical symbols as \(\text{if } \sqrt{x} = y, \text{ then } y^2 = x\). Remember that every positive number has two square roots: a positive and a negative one. However, in most practical situations, we use the positive square root.
simplifying fractions
Simplifying fractions means reducing them to their simplest form. A fraction is in its simplest form when the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1. For instance, simplifying \(\frac{4}{8}\) involves dividing both the numerator and the denominator by 4, giving \(\frac{1}{2}\). This process helps make calculations easier and results more understandable.
decimal to fraction conversion
Converting decimals to fractions can make square root calculations easier. To convert a decimal to a fraction, follow these steps:
1. Count the number of decimal places (digits to the right of the decimal point).
2. Write the decimal number as a fraction with the corresponding power of 10 as the denominator.
    Example: \(0.04 = \frac{4}{100}\)
3. Simplify the fraction to its lowest terms by dividing the numerator and the denominator by their greatest common factor (GCF).
    Example: \(\frac{4}{100}\) simplifies to \(\frac{1}{25}\)
By converting and simplifying, operations like finding square roots become more manageable.

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Most popular questions from this chapter

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt[3]{a^{2}}}{\sqrt[4]{a}}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt{a b^{3}}}{\sqrt[5]{a^{2} b^{3}}}$$

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non negativereal numbers. $$f(x)=x+\sqrt{7}, g(x)=x-\sqrt{7}$$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt{18 a^{3}} \sqrt{18 a^{3}} $$

Review simplifying rational expressions (Sections\(7.1-7.5)\) Perform the indicated operation and, if possible, simplify. $$ \frac{\frac{1}{x+1}-\frac{2}{x}}{\frac{3}{x}+\frac{1}{x+1}}[7.5] $$
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