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Chapter 8: Problem 273

Simplify. \(\frac{6}{3-\sqrt{7}}\)

Short Answer

Expert verified
9 + 3\sqrt{7}

Step by step solution

01

Identify the problem

The expression to simplify is \(\frac{6}{3 - \sqrt{7}}\). The denominator contains a square root, requiring rationalization.
02

Rationalize the denominator

To remove the square root, multiply both the numerator and the denominator by the conjugate of the denominator, which is \(3 + \sqrt{7}\). This leads to \( \frac{6}{3 - \sqrt{7}} \times \frac{3 + \sqrt{7}}{3 + \sqrt{7}} \).
03

Apply the multiplication

Multiply the numerator: \(6 \times (3 + \sqrt{7}) = 18 + 6\sqrt{7}\). Multiply the denominator: (3 - \sqrt{7})(3 + \sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9 - 7 = 2.
04

Simplify the expression

Combining the results, the simplified form becomes \(\frac{18 + 6\sqrt{7}}{2} = 9 + 3\sqrt{7}\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions means transforming them into a more manageable form without changing their value. This involves combining like terms, reducing fractions, and eliminating complex parts, such as square roots in the denominator. For instance, an expression like \( \frac{6}{3 - \sqrt{7}}\) isn't in its simplest form due to the square root in the denominator. To simplify, we need to rationalize the denominator.
Square Roots
Square roots are values that, when multiplied by themselves, give the original number. They are often represented by the symbol \( \sqrt{} \). In algebra, dealing with square roots is common, and sometimes it's necessary to remove them from the denominator of a fraction for simplification. For example, in the expression \( \frac{6}{3 - \sqrt{7}}\), the term \(-\sqrt{7}\) is in the denominator, complicating the expression.
Conjugates
Conjugates are expressions used to rationalize the denominator. The conjugate of \(a - b\) is \(a + b\), and vice versa. Multiplying a fraction by the conjugate of its denominator helps remove square roots from the denominator. Consider \( \frac{6}{3 - \sqrt{7}}\). To rationalize, you multiply by the conjugate, \(\frac{3 + \sqrt{7}}{3 + \sqrt{7}}\). This process eliminates the square root in the denominator and simplifies the expression.
Fractions
Fractions consist of a numerator and a denominator. Simplifying fractions involves reducing them to their lowest terms. In algebra, this includes handling square roots and other complex terms in the denominator. The given problem, \( \frac{6}{3 - \sqrt{7}}\), requires rationalization before simplification. Multiplying by the conjugate and simplifying the result leads to the simpler form \( 9 + 3\sqrt{7} \).

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