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Chapter 7: Problem 52

Perform the indicated operations. Assume that all variables represent positive real numbers. \(\sqrt{\frac{12}{16}}+\sqrt{\frac{48}{64}}\)

Short Answer

Expert verified
The answer is 2

Step by step solution

01

- Simplify the fractions inside the square roots

Simplify \(\frac{12}{16}\) and \(\frac{48}{64}\). \(\frac{12}{16} = \frac{3}{4}\) and \(\frac{48}{64} = \frac{3}{4}\).
02

- Replace the fractions in the square roots

Rewrite the expression using the simplified fractions: \(\frac{\frac{12}{16}} + \frac{\frac{48}{64}}\) becomes \( \frac{\frac{3}{4}} + \frac{\frac{3}{4}}\).
03

- Simplify the square roots

Simplify the square roots: \( \frac{ \frac{3}{4} } = \frac{\frac{\frac{\frac{3}{2}{2}}\)
04

- Combine like terms

Add the simplified square roots: \(\frac{3}{2})\) + \(\frac{3}{2}\text{terms}\)\
05

- Simplify the result

Combine the fractions: \( \frac{2}{1}\)

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Key Concepts

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

Square Roots
Square roots are operations that help us find a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by itself (3 * 3) equals 9. The square root is denoted by the radical symbol (鈭�).
In this exercise, we dealt with the square roots of fractions. When simplifying \(\sqrt{\frac{12}{16}}\), our first goal is to simplify the fraction inside the square root itself. This brings us to discuss fractions next.
Fractions
Fractions represent parts of a whole and can be simplified by dividing the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD). For instance, \(\frac{12}{16}\) simplifies to \(\frac{3}{4}\). Similarly, \(\frac{48}{64}\) also simplifies to \(\frac{3}{4}\), because 48 and 64 can both be divided by their GCD, which is 16.
This is a crucial step because simplifying fractions inside the square root can make the following operations much easier. After simplifying, the exercise becomes \(\sqrt{\frac{3}{4}} + \sqrt{\frac{3}{4}}\).
Simplifying Expressions
Once the fractions inside the square roots are simplified, we worked on simplifying the square roots themselves. The square root of a fraction can be expressed as the square root of the numerator over the square root of the denominator. For \(\sqrt{\frac{3}{4}}\), this becomes \(\frac{\sqrt{3}}{\sqrt{4}}\). The square root of 4 is 2, so this simplifies to \(\frac{\sqrt{3}}{2}\).
The equation then becomes \(\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}\). These terms can be combined like any other fractions: \( \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\). Hence, \(\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}\) simplifies to \( \frac{2\sqrt{3}}{2} = \sqrt{3}\) when combining the fractions.

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Most popular questions from this chapter

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[3]{x z} \cdot \sqrt{z} $$

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