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Chapter 5: Problem 15

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt{\frac{9}{36}}\)

Short Answer

Expert verified
The answer is \(\frac{1}{2}\).

Step by step solution

01

Simplify the Fraction Inside the Square Root

Start by simplifying the fraction inside the square root: \(\frac{9}{36}\). Both the numerator and the denominator have a common factor of 9. Divide the numerator by 9 to get 1, and the denominator by 9 to get 4. This simplifies the fraction to \(\frac{1}{4}\).
02

Apply Square Root to the Simplified Fraction

Apply the square root to the simplified fraction \(\frac{1}{4}\). The square root of a fraction can be taken as the square root of the numerator and the square root of the denominator separately. Thus, \(\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}}\).
03

Calculate the Square Roots

Calculate the square roots: \(\sqrt{1} = 1\) and \(\sqrt{4} = 2\). This gives \(\frac{1}{2}\) as the result of \(\frac{\sqrt{1}}{\sqrt{4}}\).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions involves reducing them to their most basic form. This is done by dividing the numerator and the denominator by their greatest common factor (GCF). In the context of the problem, the fraction inside the square root is \( \frac{9}{36} \).
  • First, identify a number that evenly divides both the numerator and the denominator. In this case, the number is 9.
  • Next, divide both the top number (9) and the bottom number (36) by this common factor. This process transforms the fraction into \( \frac{1}{4} \), a much simpler expression.
By simplifying fractions, we make calculations more manageable and understand the relationships between numbers more clearly.
Common Factors
Common factors are numbers that divide exactly into two or more bigger numbers. For finding the common factor of a fraction, look at the numerator and the denominator to see if there's a number they both can be divided by evenly.
  • In \( \frac{9}{36} \), the number 9 is a common factor because it can divide both 9 and 36 without leaving a remainder.
  • If both the numerator and the denominator have more than one common factor, choose the greatest one to simplify as much as possible.
Identifying common factors is an essential first step in simplifying fractions, which in turn can simplify other mathematical operations like taking square roots.
Square Roots of Fractions
Finding the square root of a fraction may seem complex, but it can be easier when broken down into straightforward steps. When you have a fraction under a square root, you can simplify the process by taking the square root of both the numerator and the denominator separately.
  • For the fraction \( \frac{1}{4} \), you find the square root of 1 (the numerator), which is 1.
  • Then, find the square root of 4 (the denominator), which is 2.
This gives us \( \frac{1}{2} \) as the simplified result. Basically, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Understanding this method helps in solving any similar problems involving square roots of fractions.

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Most popular questions from this chapter

Carlos's first computer had a processing speed of (1.6) \(\left(10^{6}\right)\) hertz. He recently purchased a laptop computer with a processing speed of \((1.33)\left(10^{9}\right)\) hertz. Approximately how many times faster is the processing speed of his laptop than that of his first computer? Express the result in decimal form.

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[4]{27}}{\sqrt{3}}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\frac{56 a^{\frac{1}{6}}}{8 a^{\frac{1}{4}}}\)

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(6,120,000\)

Use scientific notation and the properties of exponents to help you perform the following operations. \(\sqrt[3]{8000}\)
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