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Magazine Chapter 8: Problem 63
Rationalize each denominator. Write quotients in lowest terms. $$ \frac{\sqrt{108}}{3+3 \sqrt{3}} $$
Short Answer
Expert verified
3 - \sqrt{3}
Step by step solution
01
- Simplify the numerator
First, simplify the numerator by breaking down the square root of 108. We have: \( \sqrt{108} = \sqrt{36 \times 3} = 6 \sqrt{3} \). So the expression becomes: \( \frac{6 \sqrt{3}}{3 + 3 \sqrt{3}} \).
02
- Factor the denominator
Factor out common terms from the denominator: \( 3 + 3 \sqrt{3} = 3(1 + \sqrt{3}) \). Now the expression looks like this: \( \frac{6 \sqrt{3}}{3(1 + \sqrt{3})} \).
03
- Simplify the fraction
Since the numerator and the denominator have a common factor of 3, simplify the fraction: \( \frac{6 \sqrt{3}}{3(1 + \sqrt{3})} = \frac{6 \sqrt{3}}{3} \times \frac{1}{1 + \sqrt{3}} = 2 \sqrt{3} \times \frac{1}{1 + \sqrt{3}} = \frac{2 \sqrt{3}}{1 + \sqrt{3}} \).
04
- Rationalize the denominator
Multiply the numerator and the denominator by the conjugate of the denominator: \( 1 - \sqrt{3} \). \ \ Thus, \( \frac{2 \sqrt{3}}{1 + \sqrt{3}} \times\frac{1 - \sqrt{3}}{1 - \sqrt{3}} \).
05
- Multiply and simplify
Carry out the multiplication in both the numerator and the denominator: \( (2 \sqrt{3})(1 - \sqrt{3}) = 2 \sqrt{3} - 2 \sqrt{9} = 2 \sqrt{3} - 6 \). For the denominator: \( (1 + \sqrt{3})(1 - \sqrt{3}) = \1 - (\sqrt{3})^2 = 1 - 3 = -2 \). So we have: \( \frac{2 \sqrt{3} - 6}{-2} \).
06
- Simplify the final expression
Simplify the expression by dividing each term by -2: \( \frac{2 \sqrt{3}}{-2} - \frac{6}{-2} = - \sqrt{3} + 3 \). Therefore, the rationalized form is: \ 3 - \sqrt{3}. \
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Radicals
Radicals often show up in mathematical problems, especially under a square root symbol. Simplifying radicals makes calculations much easier.
Take for example, \(\frac{\text{the square root of 108}}{3+3 \text{ the square root of 3}}\). Simplifying starts by breaking down the radicals into their prime factors.
For \( \text{the square root of 108} \), we perform the following:
Take for example, \(\frac{\text{the square root of 108}}{3+3 \text{ the square root of 3}}\). Simplifying starts by breaking down the radicals into their prime factors.
For \( \text{the square root of 108} \), we perform the following:
- 108 = 36 脳 3
- \(\text{Square root of 36 is 6}\)
- We then get \( 6 \text{the square root of 3} \)
Conjugates
Conjugates are used to rationalize denominators in problems containing radicals. You multiply by a fraction equivalent to 1, where the denominator is its conjugate.
For instance, when we have a denominator like \( 1 + \text{ the square root of 3} \):
Let's see this in action:
The expression is \( \frac{2 \text{ the square root of 3}}{1 + \text{ the square root of 3}} \). Multiplying both by \( 1 - \text{ the square root of 3} \):
For instance, when we have a denominator like \( 1 + \text{ the square root of 3} \):
- The conjugate would be \( 1 - \text{ the square root of 3} \)
- This helps in eliminating the radical from the denominator
Let's see this in action:
The expression is \( \frac{2 \text{ the square root of 3}}{1 + \text{ the square root of 3}} \). Multiplying both by \( 1 - \text{ the square root of 3} \):
- \( \frac{2 \text{ the square root of 3}}{1 + \text{ the square root of 3}} \times \frac{1 - \text{ the square root of 3}}{1 - \text{ the square root of 3}} \)
- The denominator becomes \( (1 + \text{ the square root of 3})(1 - \text{ the square root of 3}) = 1 - (\text{ the square root of 3})^2 = 1 - 3 = -2 \)
Fractions
Fractions involve dividing one quantity by another. Simplifying fractions ensures that expressions are in their lowest terms.
Look at the fraction \( \frac{6 \text{ the square root of 3}}{3 (1 + \text{ the square root of 3})} \):
like turning \( \frac{2 \text{the square root of 3} - 6}{-2} \) into \( 3 - \text{ the square root of 3} \).
Fractions often get simplified through addition, subtraction, multiplication, or division of both the numerator and denominator by common factors.
Look at the fraction \( \frac{6 \text{ the square root of 3}}{3 (1 + \text{ the square root of 3})} \):
- Both the numerator and denominator are divisible by 3
- Dividing both by 3 simplifies it to \( \frac{2 \text{ the square root of 3}}{1 + \text{ the square root of 3}} \)
- This form is easier to handle for rationalizing the denominator
like turning \( \frac{2 \text{the square root of 3} - 6}{-2} \) into \( 3 - \text{ the square root of 3} \).
Fractions often get simplified through addition, subtraction, multiplication, or division of both the numerator and denominator by common factors.
