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Chapter 8: Problem 128

Rationalize the numerator of \(\frac{\sqrt{5}+2}{5}\).

Short Answer

Expert verified
\(\frac{1}{5(\sqrt{5}-2)}\)

Step by step solution

01

Identify the conjugate of the numerator

The conjugate of the numerator \(\sqrt{5}+2\) is found by changing the sign in the middle of the two terms. So, the conjugate is \(\sqrt{5}-2\).
02

Multiply the fraction by the conjugate

Multiply both the numerator and the denominator of the fraction by the conjugate of the numerator. This gives: \(\frac{(\sqrt{5}+2)(\sqrt{5}-2)}{5(\sqrt{5}-2)}\).
03

Simplify the expression

Simplify the numerator using the difference of squares, which states that \(a^2 - b^2 = (a+b)(a-b)\). Using this formula, the numerator becomes 5 - 4 = 1. Similarly, simplify the denominator: 5(\(\sqrt{5}-2\)). The final expression is then: \(\frac{1}{5(\sqrt{5}-2)}\).

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Key Concepts

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

Understanding Conjugates
Conjugates are essential in mathematics, especially when dealing with irrational numbers like square roots. When you have a binomial expression involving a square root, such as \( \sqrt{5} + 2 \), the conjugate is formed by changing the sign between the terms. Here, it becomes \( \sqrt{5} - 2 \).

This technique is useful for eliminating square roots from the numerator of a fraction, making it 'rational'.

To rationalize, you multiply both the numerator and denominator by this conjugate:
  • Preserves the equality of the fraction
  • Helps to simplify expressions by reducing irrational components
The idea is that any irrational part in the numerator will cancel out, thanks to their differences being squared.
Exploring the Difference of Squares
The difference of squares is a foundational algebraic identity: \( a^2 - b^2 = (a+b)(a-b) \). It's particularly handy when working with conjugates, allowing us to simplify expressions.

For instance, when you multiply \((\sqrt{5}+2)(\sqrt{5}-2)\), you apply this identity. Here, \( a = \sqrt{5} \) and \( b = 2 \). Thus:

\[ (\sqrt{5})^2 - 2^2 = 5 - 4 = 1 \]

This simplification results because the middle terms in the expansion cancel each other out. The difference of squares helps transform complex expressions into more manageable forms.
Simplifying Expressions
Simplifying expressions is about reducing them to their simplest form. Once the difference of squares is applied, any irrational numbers can be removed from the numerator.

After rationalizing by the conjugate, the expression: \( \frac{(\sqrt{5}+2)(\sqrt{5}-2)}{5(\sqrt{5}-2)} \) simplifies to:
  • Numerator: 1, after applying difference of squares
  • Denominator: \( 5(\sqrt{5}-2) \), keeping the structure of the original expression
The final expression \( \frac{1}{5(\sqrt{5}-2)} \) shows a fully rationalized and simplified form.

This process makes understanding and using the expression easier, as it is now free of irrational numbers in the numerator.

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Most popular questions from this chapter

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{720}$$

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