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Chapter 0: Problem 28

Simplify each expression involving square roots. $$ \sqrt{\frac{3}{16}} $$

Short Answer

Expert verified
\( \frac{\sqrt{3}}{4} \)

Step by step solution

01

Rewrite the square root of the fraction

The square root of a fraction can be rewritten as the square root of the numerator divided by the square root of the denominator. For the given problem, rewrite \( \sqrt{\frac{3}{16}} \) as \( \frac{\sqrt{3}}{\sqrt{16}} \).
02

Simplify the denominator

Determine the square root of the denominator. Since the denominator is 16, \( \sqrt{16} = 4 \). Now the expression is \( \frac{\sqrt{3}}{4} \).

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Key Concepts

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

Square Root of a Fraction
Taking the square root of a fraction may seem tricky, but it's quite simple if you follow a few basic steps. When you encounter a fraction inside a square root, like \[ \sqrt{\frac{3}{16}} \], you can split this into two parts: the square root of the numerator and the square root of the denominator. This is based on the property of square roots, which states that \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]. In our example, this means we can rewrite the expression as \[ \frac{\sqrt{3}}{\sqrt{16}} \]. Breaking down the fraction this way makes the problem more manageable.
Numerator and Denominator
The numerator and denominator are vital parts of any fraction. The numerator is the number above the fraction line, and the denominator is the number below it. In the fraction \[ \frac{3}{16} \], the 3 is the numerator, and the 16 is the denominator. When you simplify the square root of a fraction, you need to take the square root of both the numerator and the denominator separately. Remember, in our example, we have \[ \sqrt{3} \] and \[ \sqrt{16} \].
Simplifying Expressions
Simplifying expressions involving square roots requires patience and accuracy. After splitting the square root of a fraction, focus on simplifying each part individually. For example, \[ \frac{\sqrt{3}}{\sqrt{16}} \] can be broken down further. The square root of 16 is 4 since \[ 4 \times 4 = 16 \]. This means our expression simplifies to \[ \frac{\sqrt{3}}{4} \]. At this stage, the numerator, \[ \sqrt{3} \], cannot be simplified further since 3 is not a perfect square. Finally, ensure your simplified expression is in its most reduced form. Hence, \[ \frac{\sqrt{3}}{4} \] is the simplest form of \[ \sqrt{\frac{3}{16}} \].

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Most popular questions from this chapter

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(a) $$

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=\sqrt{x}+30 $$

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ g(b) $$

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ \frac{f(1)}{g(1)} $$

Explain why \((f \circ g)^{-1}=g^{-1} \circ f^{-1}\) for any invertible functions \(f\) and \(g .\) Discuss any restrictions on the domains and ranges of \(f\) and \(g\) for this equation to be correct.
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