/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 Simplify each radical. $$ \f... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 76

Simplify each radical. $$ \frac{\sqrt{2}+\sqrt{7}}{\sqrt{5}+\sqrt{3}} $$

Short Answer

Expert verified
\(\frac{\sqrt{10} - \sqrt{6} + \sqrt{35} - \sqrt{21}}{2}\)

Step by step solution

01

- Rationalize the denominator

To simplify the expression \(\frac{\sqrt{2}+\sqrt{7}}{\sqrt{5}+\sqrt{3}}\), first multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{5}+\sqrt{3}\) is \(\sqrt{5}-\sqrt{3}\). Thus, multiply by \(\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}\).
02

- Apply distribution in the numerator and denominator

Distribute in both the numerator and the denominator: \(\frac{(\sqrt{2} + \sqrt{7})(\sqrt{5} - \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})}\).
03

- Simplify the denominator using difference of squares

The denominator simplifies using the difference of squares formula: \( (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \). So the new denominator is 2.
04

- Expand and simplify the numerator

For the numerator, you need to expand and combine like terms. First, distribute \(\sqrt{2}(\sqrt{5} - \sqrt{3}) = \sqrt{2\cdot5} - \sqrt{2\cdot3} = \sqrt{10} - \sqrt{6}\). Then distribute \(\sqrt{7}(\sqrt{5} - \sqrt{3}) = \sqrt{7\cdot5} - \sqrt{7\cdot3} = \sqrt{35} - \sqrt{21}\). Therefore, the expanded numerator is \(\sqrt{10} - \sqrt{6} + \sqrt{35} - \sqrt{21}\).
05

- Combine the fractions

Putting it all together, the simplified form of the original expression is: \(\frac{\sqrt{10} - \sqrt{6} + \sqrt{35} - \sqrt{21}}{2}\).

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Key Concepts

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

Rationalizing the Denominator
Rationalizing the denominator is a method used to eliminate radicals from the denominator of a fraction. This makes the expression easier to understand and work with.

Here's how it works:
  • Identify the radical in the denominator.
  • Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate is formed by changing the sign between two terms. For example, the conjugate of \(a + b\) is \(a - b\).
In our example, the denominator \(\sqrt{5} + \sqrt{3}\) has the conjugate as \(\sqrt{5} - \sqrt{3}\). Multiplying by \(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}\) keeps the value same but changes the form into something more manageable.
Conjugate Multiplication
Conjugate multiplication is a powerful technique to simplify expressions with radicals. When you multiply a term by its conjugate, you employ the difference of squares formula.

Let's see the steps:
  • Take the original expression and the conjugate of the denominator.
  • Multiply both the numerator and denominator by the conjugate.
  • In the case of \(\frac{\sqrt{2} + \sqrt{7}}{\sqrt{5} + \sqrt{3}}\), we multiply by \(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}\).
The multiplication in the denominator \( (\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) \) uses the difference of squares to become \ 5 - 3 = 2.\ For the numerator, we distribute each term such as ... and then combine them.
Difference of Squares
The difference of squares is a specific algebraic formula used to simplify expressions. It states that \[a^2 - b^2 = (a + b)(a - b)\].

Here's how to apply it:
  • Recognize the form \[a^2 - b^2 = (a + b)(a - b)\]
  • Use the formula to simplify products of conjugates.
In the provided solution, the denominator becomes \( (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \). This greatly simplifies our result.
Using this makes complex radicals easier to manage and understand.

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