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Chapter 8: Problem 17

Simplify. $$\sqrt{\frac{3 p}{64}}$$

Short Answer

Expert verified
\(\frac{\sqrt{3p}}{8}\)

Step by step solution

01

Separate the Square Root

Use the property of square roots that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). So, we get \(\sqrt{\frac{3p}{64}} = \frac{\sqrt{3p}}{\sqrt{64}}\).
02

Simplify the Denominator

\(\sqrt{64} = 8\) since 64 is a perfect square. Therefore, \(\frac{\sqrt{3p}}{\sqrt{64}} = \frac{\sqrt{3p}}{8}\).
03

Combine the Fraction

The simplified form is \(\frac{\sqrt{3p}}{8}\). Since nothing further can be simplified, the final answer is \(\frac{\sqrt{3p}}{8}\).

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Key Concepts

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

Simplifying Square Roots
Square roots allow us to find a number which, when multiplied by itself, gives the original number. The square root of 9, for example, is 3 because 3 脳 3 = 9. When dealing with square roots in expressions, we'll often encounter them with fractions and variables.
In our exercise, we started with the expression \(\frac{\text{3p}}{64}\). We used the property of square roots that states: \(\frac{\text{a}}{\text{b}} = \frac{\text{鈭歛}}{\text{鈭歜}}\). This step helps break down complex expressions into simpler components.
Fraction Simplification with Square Roots
In the fraction \(\frac{\text{3p}}{64}\), both the numerator and the denominator can be dealt with separately. Using the property \(\frac{\text{鈭歛}}{\text{鈭歜}}\), we transformed the expression into \(\frac{\text{鈭�(3p)}}{\text{鈭�64}}\).
This property is particularly useful because it allows you to handle each part independently. For the denominator, since 64 is a perfect square, we can simplify \(\text{鈭�64}\) to 8, converting our fraction directly into \(\frac{\text{鈭�(3p)}}{8}\). Breaking complex fractions this way makes them more manageable.
Understanding Perfect Squares
Perfect squares are numbers that have a whole number as their square root. Examples include 1, 4, 9, 16, 25, and 64. Recognizing these numbers can really speed up the process of simplifying square roots.
In our exercise, we noted that 64 is a perfect square because \(\text{8} 脳 \text{8} = \text{64}\). Knowing this allowed us to simplify \(\frac{\text{鈭�(3p)}}{\text{鈭�64}}\) into \(\frac{\text{鈭�(3p)}}{8}\). When simplifying expressions, always look out for perfect squares. They make the process quicker and easier.

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