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Chapter 1: Problem 6

Express each in terms of the simplest possible radical. $$\sqrt{\frac{15}{20}}$$

Short Answer

Expert verified
\(\sqrt{\frac{15}{20}} = \frac{\sqrt{3}}{2}\).

Step by step solution

01

Simplify the fraction inside the square root

Begin by simplifying the fraction inside the square root. The fraction is \(\frac{15}{20}\). Find the greatest common divisor (GCD) of 15 and 20, which is 5. Simplify the fraction by dividing both the numerator and the denominator by 5: \(\frac{15}{20} = \frac{3}{4}\).
02

Apply square root to the simplified fraction

Now apply the square root to the simplified fraction \(\frac{3}{4}\). Since the square root of a fraction can be expressed as the square root of the numerator divided by the square root of the denominator, we have: \(\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}}\).
03

Simplify the expression further

Recognize that \(\sqrt{4} = 2\). Substitute this into the expression, resulting in \(\frac{\sqrt{3}}{2}\). Now, the expression \(\frac{\sqrt{3}}{2}\) is in its simplest radical form.

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Key Concepts

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

Greatest Common Divisor
Finding the Greatest Common Divisor (GCD) is an essential step in simplifying fractions. The GCD of two numbers is the largest number that can exactly divide both of them without leaving a remainder. To find the GCD, you can list out the factors of each number and choose the largest factor that appears in both lists. For example, when simplifying the fraction \(\frac{15}{20}\), we find that the GCD of 15 and 20 is 5. By dividing both the numerator (15) and the denominator (20) by their GCD, we simplify the fraction to \(\frac{3}{4}\). This step is crucial as it reduces the fraction to its lowest terms, making further calculations much easier and less error-prone.
Square Root of Fractions
The square root of a fraction follows a straightforward rule: the square root of a fraction \(\frac{a}{b}\) is equal to the square root of the numerator \(a\) divided by the square root of the denominator \(b\). This can be expressed as:
  • \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
Applying this rule in practice requires simplifying both the numerator and denominator, if possible. In our example, after simplifying \(\frac{15}{20}\) to \(\frac{3}{4}\), we calculate the square roots to get \(\sqrt{3}\) and \(\sqrt{4}\). Since \(\sqrt{4} = 2\), the expression becomes \(\frac{\sqrt{3}}{2}\). This method allows us to handle the root operation separately on the top and bottom parts of the fraction.
Simplifying Fractions
Simplifying fractions is the process of reducing them to their simplest form where the numerator and the denominator have no common factors other than 1. This is achieved by dividing both parts of the fraction by their greatest common divisor (GCD). This step is important because it often makes subsequent calculations, such as finding square roots, more manageable.
  • For \(\frac{15}{20}\), we simplify it to \(\frac{3}{4}\) by identifying 5 as the GCD.
  • This approach not only helps in solving mathematical problems but also improves the visual clarity of numbers, making them easier to understand and work with.
By ensuring fractions are in their simplest form, one avoids potential errors in calculation, especially when entering subsequent operations like multiplication, division, or finding roots.

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Most popular questions from this chapter

Write an inequality to represent the given interval and state whether the interval is closed, open or half-open. Also state whether the interval is bounded or unbounded. $$[-5,3]$$

Perform the indicated operation and simplify. $$\frac{3}{x-2}+\frac{5}{2-x}$$

Determine whether each statement is true for all real numbers \(x\). If the statement is false, then indicate one counterexample, i.e. a value of \(x\) for which the statement is false. $$x^{2} \geqslant x$$

Determine whether each statement is true for all real numbers \(x\). If the statement is false, then indicate one counterexample, i.e. a value of \(x\) for which the statement is false. $$x^{3}+1>x^{3}$$

Use scientific notation and the laws of exponents to perform the indicated operations. Give the result in scientific notation rounded to two significant figures. $$\left(2 \times 10^{3}\right)^{4}\left(3.5 \times 10^{5}\right)$$
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