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Chapter 10: Problem 76

Reduce to lowest terms. $$\frac{10-\sqrt{24}}{12}$$

Short Answer

Expert verified
\(\frac{5-\sqrt{6}}{6}\)

Step by step solution

01

Simplify the Radical

First simplify the square root in the numerator: \(\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}\). Thus, the expression becomes \(\frac{10-2\sqrt{6}}{12}\).
02

Find the Greatest Common Divisor (GCD)

Determine the GCD of the terms in the numerator and the denominator. Here, the terms are 10, -2\(\sqrt{6}\), and 12. The GCD of 10, 2, and 12 is 2.
03

Divide by the GCD

Divide both the numerator and the denominator by the GCD found in Step 2. This results in: \(\frac{10-2\sqrt{6}}{12} = \frac{10-2\sqrt{6}}{2 \cdot 6} = \frac{(10-2\sqrt{6})/2}{12/2} = \frac{5-\sqrt{6}}{6}\).

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Key Concepts

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

Algebraic Fractions
Algebraic fractions are fractions where both the numerator and the denominator are algebraic expressions. These fractions can include variables (like x and y), constants (like 5 or -3), and radicals (like \( \sqrt{6} \)). To simplify algebraic fractions, we can perform similar steps as we do with numerical fractions. This can include simplifying the expressions, finding common factors, and reducing them to their lowest terms.
Greatest Common Divisor
The Greatest Common Divisor (GCD) is the largest positive integer that divides all the given numbers in the expression without leaving a remainder. To find the GCD of algebraic expressions, we need to look at the numerical coefficients and find their GCD. In our exercise, the terms we need to consider for the GCD are 10, -2\( \sqrt{6} \), and 12. We ignore the radical part because it does not affect the GCD process. The GCD of 10, 2, and 12 is 2. Once we have the GCD, we use it to simplify the algebraic fraction.
Reducing Expressions
Reducing expressions means simplifying them to their most basic form. For algebraic fractions, this often involves factoring out the GCD from both the numerator and the denominator. In our example, after finding the GCD of 10 and -2\( \sqrt{6} \) is 2, we divide both the numerator and the denominator by this GCD. This process helps in minimizing the expression, making it simpler and easier to work with. After reducing, we obtain the simplest form of the fraction: \( \frac{5-\sqrt{6}}{6} \).

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Most popular questions from this chapter

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\frac{\sqrt{12}}{5}+\frac{5}{\sqrt{3}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{\frac{2}{7}}+\sqrt{\frac{7}{2}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{45 x}-\sqrt{20 x}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{18}+\sqrt{27}+\sqrt{45}$$

Factor the given expression as completely as possible. $$a^{2}+6 a-40$$
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