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Chapter 1: Problem 46

Find the following roots. See Examples 7 and \(8 .\) $$ -\sqrt{\frac{4}{25}} $$

Short Answer

Expert verified
The root is \(-\frac{2}{5}\).

Step by step solution

01

Identify the Expression

The given expression is \(-\sqrt{\frac{4}{25}}\). We are required to find the square root of \(\frac{4}{25}\) first, and then apply the negative sign.
02

Simplify the Fraction

The fraction is \(\frac{4}{25}\). This represents the division of two perfect squares, where \(4 = 2^2\) and \(25 = 5^2\).
03

Take the Square Root

The square root of a fraction \(\frac{a}{b}\) is \(\frac{\sqrt{a}}{\sqrt{b}}\), so: \[ \sqrt{\frac{4}{25}} = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5}. \]
04

Apply the Negative Sign

Now, apply the negative sign to the square root value obtained: \[-\sqrt{\frac{4}{25}} = -\frac{2}{5}.\]

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is one of the fundamental concepts in mathematics. It involves reducing fractions to their simplest form. This means using the smallest possible numerator and denominator that have the same value as the original fraction.
  • To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that divides both numbers without leaving a remainder.
  • After finding the GCD, divide both the numerator and the denominator by this number.
For example, the fraction \(\frac{8}{12}\) can be simplified. The GCD of 8 and 12 is 4. So, we divide both by 4, resulting in \(\frac{2}{3}\). Simplifying fractions helps in making calculations easier and more comprehensible, especially when you are working with complex expressions.
Square Roots
A square root is a value that, when multiplied by itself, gives the original number. In simpler terms, it is the inverse operation of squaring a number.
  • The square root of a number \(n\) is written as \(\sqrt{n}\).
  • For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
An essential property of square roots is that they apply to both whole numbers and fractions. When dealing with fractions, you can find the square root of each the numerator and the denominator separately, like in \(\sqrt{\frac{4}{25}} = \frac{2}{5}\). This makes working with radical expressions straightforward by breaking down the problem into smaller, more manageable pieces.
Perfect Squares
Perfect squares are numbers that are the product of an integer multiplied by itself. These serve as a special case in the realm of numbers and are particularly useful when dealing with square roots.
  • A perfect square always has a positive integer as its square root. For instance, 16 is a perfect square because \(4 \times 4 = 16\).
  • Recognizing perfect squares can help in simplifying expressions, especially those involving radicals.
In the context of fractions, perfect squares simplify both the numerator and the denominator, making square roots much easier to manage. For the fraction \(\frac{4}{25}\), both 4 and 25 are perfect squares, since \(2^2 = 4\) and \(5^2 = 25\). Using perfect squares allows us to take square roots easily and accurately, aiding in simplifying complex expressions efficiently.

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Most popular questions from this chapter

Simplify each expression. $$ 6.5 y-4.4(1.8 x-3.3)+10.95 $$

Choose the fraction (s) equivalent to the given fraction. (There may be more than one correct choice.) $$ \begin{aligned} \frac{(y+z)}{3 y} & \text { a. } \frac{-(y+z)}{3 y} \text { b. } \frac{-(y+z)}{-3 y} \\ & \text { c. } \frac{(y+z)}{3 y} \quad \text { d. } \frac{(y+z)}{-3 y} \end{aligned} $$

Translating Write each phrase as an algebraic expression. Use the variable \(x\) to represent each unknown number. See Example 8 . The quotient of four and the sum of a number and one.

Translating Write each phrase as an algebraic expression. Use the variable \(x\) to represent each unknown number. See Example 8 . Four, subtracted from three times a number.

Write each of the following as an algebraic expression. See Examples 12 and 13. If \(2 x\) is an even integer, represent the next even integer as an expression in \(x\).
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