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Chapter 8: Problem 58

Rationalize each denominator. Write quotients in lowest terms. $$ \frac{3 \sqrt{2}-4}{\sqrt{3}+2} $$

Short Answer

Expert verified
-3\sqrt{6} + 6\sqrt{2} + 4\sqrt{3} - 8

Step by step solution

01

Identify the Conjugate

The first step to rationalize the denominator is to identify the conjugate of the denominator. The conjugate of \( \sqrt{3} + 2 \) is \( \sqrt{3} - 2 \).
02

Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate: \[ \frac{3 \sqrt{2} - 4}{\sqrt{3} + 2} \times \frac{\sqrt{3} - 2}{\sqrt{3} - 2} \]
03

Apply the Distributive Property in the Numerator

Apply the distributive property to the numerator: \[ (3\sqrt{2} - 4)(\sqrt{3} - 2) = 3\sqrt{2} \cdot \sqrt{3} - 3\sqrt{2} \cdot 2 - 4 \cdot \sqrt{3} + 4 \cdot 2 \] Simplify: \[ 3\sqrt{6} - 6\sqrt{2} - 4\sqrt{3} + 8 \]
04

Apply the Difference of Squares in the Denominator

Use the difference of squares formula to simplify the denominator: \[ (\sqrt{3} + 2)(\sqrt{3} - 2) = (\sqrt{3})^2 - (2)^2 = 3 - 4 = -1 \]
05

Combine Numerator and Denominator

Combine the simplified numerator and denominator: \[ \frac{3\sqrt{6} - 6\sqrt{2} - 4\sqrt{3} + 8}{-1} = - (3\sqrt{6} - 6\sqrt{2} - 4\sqrt{3} + 8) = -3\sqrt{6} + 6\sqrt{2} + 4\sqrt{3} - 8 \]

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Key Concepts

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

conjugate
The first key concept is the *conjugate*. When we have a binomial involving a square root in the denominator, we use the conjugate to help remove the radical.

The conjugate of a binomial is another binomial with the same terms but with the opposite sign between them.

For example, in our problem, the conjugate of \( \sqrt{3} + 2 \) is \( \sqrt{3} - 2 \).

Multiplying by the conjugate helps since it simplifies into a rational number. This approach is fundamental for rationalizing denominators.
distributive property
Next, we apply the *distributive property*. This property states that \( a(b + c) = ab + ac \).

In our example, we distribute each term in the numerator by each term in the conjugate of the denominator: \( (3\sqrt{2} - 4)(\sqrt{3} - 2) = 3\sqrt{2} \cdot \sqrt{3} - 3\sqrt{2} \cdot 2 - 4 \cdot \sqrt{3} + 4 \cdot 2 \).

This results in simpler terms that can be combined to make further simplifications easier: \( 3\sqrt{6} - 6\sqrt{2} - 4\sqrt{3} + 8 \).
difference of squares
We now use the *difference of squares* identity, which is very useful when dealing with conjugates.

The difference of squares formula is \( (a + b)(a - b) = a^2 - b^2 \).

Using this formula on the denominator: \( (\sqrt{3} + 2)(\sqrt{3} - 2) = (\sqrt{3})^2 - (2)^2 \).

Simplifying this gives: \( 3 - 4 = -1 \).

This effectively removes the radical in the denominator making further simplification straightforward.
simplifying radicals
The last key concept is *simplifying radicals*.

After using the distributive property and the difference of squares, we further simplify our expression.

Combining the simplified numerators and denominators: \( \frac{3\sqrt{6} - 6\sqrt{2} - 4\sqrt{3} + 8}{-1} \).

This changes the problem to: \( - (3\sqrt{6} - 6\sqrt{2} - 4\sqrt{3} + 8) = -3\sqrt{6} + 6\sqrt{2} + 4\sqrt{3} - 8 \).

This simplification achieves a final rationalized form.

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Most popular questions from this chapter

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