Question

Rewrite each expression by rationalizing the denominator. $$ \frac{\sqrt{5}}{5-\sqrt{5}} $$

Short answer

Question: Rewrite the given expression with a rational denominator: $\frac{\sqrt{5}}{5-\sqrt{5}}$ Answer: $\frac{5\sqrt{5}+5}{20}$

Step-by-step solution

Identify the conjugate

The conjugate of the denominator (5 - √5) is 5 + √5.

Multiply by the conjugate

Multiply both the numerator and the denominator of the expression by the conjugate (5 + √5). $$ \frac{\sqrt{5}}{5-\sqrt{5}} \times \frac{5+\sqrt{5}}{5+\sqrt{5}} $$

Multiply the numerators

Multiply the numerators together. $$ \frac{\sqrt{5}(5+\sqrt{5})}{(5-\sqrt{5})(5+\sqrt{5})} $$

Multiply the denominators and simplify

Multiply the denominators together and simplify using the difference of squares formula, which states that (a - b)(a + b) = a^2 - b^2. $$ \frac{\sqrt{5}(5+\sqrt{5})}{(5^2-(\sqrt{5})^2)} $$

Simplify the expression

Now, simplify the expression. $$ \frac{\sqrt{5}(5+\sqrt{5})}{25-5} $$ $$ \frac{\sqrt{5}(5+\sqrt{5})}{20} $$

Distribute the square root of 5

Lastly, distribute √5 to each term in the parenthesis. $$ \frac{5\sqrt{5}+\sqrt{5}\times\sqrt{5}}{20} $$ $$ \frac{5\sqrt{5}+5}{20} $$ Now we have successfully rationalized the denominator and simplified the expression: $$ \frac{5\sqrt{5}+5}{20} $$

Key concepts

Conjugate

In the world of mathematics, the term 'conjugate' typically refers to a paired term designed to eliminate a square root or other irrational number from a denominator. In this case, the conjugate of a binomial expression like \( 5 - \sqrt{5} \) is \( 5 + \sqrt{5} \).

By multiplying both the numerator and denominator of a fraction by the conjugate of the denominator, you effectively remove any irrational components, making division and other computations much more manageable.

To apply this in practice:

• First, identify the expression in the denominator, for instance, \( 5 - \sqrt{5} \).
• The conjugate would be \( 5 + \sqrt{5} \). Notice that you change the sign in between the terms.
• Multiplying by the conjugate helps rationalize the denominator, turning it into a rational number.

In rationalizing denominators, using the conjugate ensures the elimination of surd or irrational numbers.

Difference of Squares

The difference of squares is a notable algebraic pattern that helps in simplification when working with conjugates. It's expressed as \((a - b)(a + b) = a^2 - b^2\). This rule is instrumental when dealing with expressions involving conjugates.

When you multiply the denominator \((5 - \sqrt{5})(5 + \sqrt{5})\), it perfectly fits the difference of squares formula:

\(a = 5\) and \(b = \sqrt{5}\)
Substitute these values into the formula:

• \(a^2 = 5^2 = 25\)
• \(b^2 = (\sqrt{5})^2 = 5\)
• Subtract the values: \( 25 - 5 = 20 \)

Now, the denominator is simplified to a simple number, which is entirely rational. This process is at the heart of rationalizing the denominator, as it converts an irrational denominator into a more manageable form.

Simplifying Expressions

Simplifying expressions takes several steps, especially when they involve radicals and irrational numbers. After multiplying by the conjugate and applying difference of squares, it is crucial to simplify further to get the expression into its neatest form.

Once the multiplication has been carried out, such as \( \frac{\sqrt{5}(5+\sqrt{5})}{20} \), distributing and simplifying can begin.

• Distribute \(\sqrt{5}\) throughout the numerator:
• \(5\sqrt{5} + \sqrt{5} \times \sqrt{5} = 5\sqrt{5} + 5\)

This makes the expression:
\(\frac{5\sqrt{5} + 5}{20}\)

To simplify further, factor out common terms when possible, making the expression less complex. This step-by-step process illuminates how seemingly complicated expressions can be reduced and managed more easily.