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Chapter 10: Problem 67

Rationalize each denominator. $$\frac{\sqrt{7}-\sqrt{3}}{\sqrt{3}-\sqrt{7}}$$

Short Answer

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Step by step solution

01

- Identify the Conjugate

To rationalize the denominator, identify the conjugate of the denominator. The conjugate of \(\sqrt{3} - \sqrt{7}\) is \(\sqrt{3} + \sqrt{7}\).
02

- Multiply Numerator and Denominator by the Conjugate

Multiply the fraction \(\frac{\sqrt{7} - \sqrt{3}}{\sqrt{3} - \sqrt{7}}\) by \(\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} + \sqrt{7}}\). This gives: \[ \frac{(\sqrt{7} - \sqrt{3})(\sqrt{3} + \sqrt{7})}{(\sqrt{3} - \sqrt{7})(\sqrt{3} + \sqrt{7})} \]
03

- Expand the Numerator

Expand the numerator \( (\sqrt{7} - \sqrt{3})(\sqrt{3} + \sqrt{7}) \) using the distributive property: \(\sqrt{7} \cdot \sqrt{3} + \sqrt{7} \cdot \sqrt{7} - \sqrt{3} \cdot \sqrt{3} - \sqrt{3} \cdot \sqrt{7}\). This simplifies to: \(\sqrt{21} + 7 - 3 - \sqrt{21} = 4\).
04

- Simplify the Denominator

Use the difference of squares formula in the denominator \( (\sqrt{3} - \sqrt{7})(\sqrt{3} + \sqrt{7}) = (\sqrt{3})^2 - (\sqrt{7})^2 = 3 - 7 = -4 \).
05

- Simplify the Fraction

Now, put the results together: \[ \frac{4}{-4} = -1 \].

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Key Concepts

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

Conjugate
To rationalize a denominator, we use the concept of a conjugate. A conjugate is a pair of expressions with the same terms but opposite signs. For example, the conjugate of \(\sqrt{3} - \sqrt{7}\) is \(\sqrt{3} + \sqrt{7}\). We use the conjugate to eliminate square roots from the denominator. By multiplying both the numerator and the denominator by the conjugate, the troublesome square roots cancel out, leaving a rational number in the denominator. This process helps simplify expressions and is crucial for more advanced algebraic manipulations.
Distributive Property
The distributive property is a fundamental algebraic principle. It states that a term outside a parenthesis can be multiplied with each term inside the parenthesis. When we multiply the numerator in \(\frac{(\sqrt{7} - \sqrt{3})(\sqrt{3} + \sqrt{7})}\), we apply the distributive property like this:
  • First, multiply \(\sqrt{7}\) with each term inside the second parentheses: \(\sqrt{7} \cdot \sqrt{3} = \sqrt{21}\) and \(\sqrt{7} \cdot \sqrt{7} = 7\).
  • Next, multiply \(\-\sqrt{3}\) with each term of the conjugate: \(\-\sqrt{3} \cdot \sqrt{3} = -3\) and \(\-\sqrt{3} \cdot \sqrt{7} = -\sqrt{21}\).
By simplifying the expression, we cancel out the \(\sqrt{21}\) terms, leaving just \(7 - 3 = 4\).
Difference of Squares
The difference of squares formula is used when you multiply the sum and the difference of the same two terms. This formula is: \(a^2 - b^2 = (a+b)(a-b)\). Applying this to \(\frac{((\sqrt{3} - \sqrt{7})(\sqrt{3} + \sqrt{7}))}\):
- Observe that \(a = \sqrt{3}\) and \(b = \sqrt{7}\).
  • Apply the formula: \(\sqrt{3}^2 - \sqrt{7}^2 = 3 - 7 = -4\).

    • This reduces the denominator to a simple number. Using the difference of squares is a powerful tool for simplifying the rationalization process and other algebraic expressions.

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    Most popular questions from this chapter

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