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Chapter 8: Problem 45

Rationalize each denominator. Write quotients in lowest terms. $$ \frac{1}{2+\sqrt{5}} $$

Short Answer

Expert verified
The rationalized fraction is \(-2 + \sqrt{5}\).

Step by step solution

01

Identify the Conjugate

To rationalize the denominator, identify the conjugate of the denominator. The conjugate of the denominator \(2 + \sqrt{5}\) is \(2 - \sqrt{5}\).
02

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \(2 - \sqrt{5}\): \[ \frac{1}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} \].
03

Simplify the Numerator

Distribute 1 in the numerator: \[ \frac{2 - \sqrt{5}}{(2 + \sqrt{5})(2 - \sqrt{5})} \].
04

Simplify the Denominator

Use the difference of squares formula: \[ (a + b)(a - b) = a^2 - b^2 \. Therefore, (2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1 \].
05

Write the Final Expression

Substitute the simplified denominator back in: \[ \frac{2 - \sqrt{5}}{-1} = -2 + \sqrt{5} \].

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Key Concepts

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

Conjugate
Understanding the concept of the 'conjugate' is crucial when rationalizing denominators. The conjugate of a binomial expression involves changing the sign between two terms. For example, the conjugate of \(2 + \sqrt{5} \) is \( 2 - \sqrt{5} \). Always remember:
  • The conjugate flips the sign in the middle of the binomial.
  • It helps eliminate the square root when multiplied together.
The reason we use the conjugate is that it helps to simplify the denominator without changing the value of the expression. This process is particularly useful for algebraic expressions involving radicals. By choosing the right conjugate, we can get rid of those pesky square roots in the denominator.
Difference of Squares
The 'difference of squares' formula is a mathematical tool that makes simplifying expressions easier. It states: \[ (a + b)(a - b) = a^2 - b^2 \]. In our example, when we multiply \(2 + \sqrt{5}\) by its conjugate \(2 - \sqrt{5} \), we apply this formula:
  • \(a \) is 2 and \( b \) is \( \sqrt{5} \).
  • Perform the multiplication: \( (2+\sqrt{5})(2 - \sqrt{5})\).
  • Use \( a^2 - b^2\) to simplify: \(2^2 - (\sqrt{5})^2 = 4 - 5 = -1\).
So, \( (2+\sqrt{5})(2 - \sqrt{5})=-1\). This step shows how the denominator becomes simpler when we use the difference of squares formula. By understanding this concept, we make rationalizing denominators a piece of cake!
Simplification
Simplification is the last and crucial step in rationalizing a denominator. After using the conjugate and the difference of squares formula, we end up with a more straightforward expression. Here鈥檚 how:
  • The numerator becomes \(2 - \sqrt{5}\) after multiplication.
  • The denominator simplifies to -1.
  • Our expression is now \(\frac{2 - \sqrt{5}}{-1}\).
Next, we just divide each part of the numerator by the simplified denominator:
\ \(\frac{2 - \sqrt{5}}{-1} = -2 + \sqrt{5}\).
Lastly, always check to ensure the expression is in its simplest form. Here, \(-2 + \sqrt{5}\) is the final, simplified result. Understanding these steps makes it easier for you to handle more complex algebraic problems in the future!

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