Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{5}{\sqrt{2}-1}$$

Short answer

5\sqrt{2} + 5

Step-by-step solution

Identify the Conjugate of the Denominator

To rationalize the denominator of the expression, identify the conjugate of the denominator. The denominator is \(\sqrt{2} - 1\). The conjugate of \(\sqrt{2} - 1\) is \(\sqrt{2} + 1\).

Multiply the Numerator and Denominator by the Conjugate

Multiply both the numerator and denominator by \(\sqrt{2} + 1\). This gives: \[ \frac{5}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1} \]

Apply Distribution in the Numerator

Distribute in the numerator: \(5 \cdot (\sqrt{2} + 1) = 5\sqrt{2} + 5\)

Apply Difference of Squares in the Denominator

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

Combine Both Results

Put together the simplified numerator and denominator: \[ \frac{5\sqrt{2} + 5}{1} = 5\sqrt{2} + 5 \]

Key concepts

Conjugate

When rationalizing the denominator, we often encounter square roots or other irrational numbers in the denominator. To eliminate these, we use the concept of the 'conjugate.' The conjugate of a binomial expression like \(a - b\) is \(a + b\).
In our example, the denominator is \(\sqrt{2} - 1\), so the conjugate is \(\sqrt{2} + 1\).
This method works because multiplying a binomial by its conjugate results in a difference of squares, which removes the irrational parts.

Difference of Squares

The difference of squares formula is a powerful algebraic identity: \[ a^2 - b^2 = (a+b)(a-b). \] When applied to conjugates such as \(\sqrt{2} - 1 \) and \(\sqrt{2} + 1 \), it simplifies to: \[(\sqrt{2})^2 - (1)^2 = 2 - 1 = 1.\]
This helps in removing square roots from the denominator.
In this example, we saw that multiplying \(\sqrt{2} - 1\) by \(\sqrt{2} + 1\) gives us 1, which is a rational number.

Simplifying Expressions

When rationalizing expressions, we also need to simplify the numerator. This involves distributing any factors or performing multiplication.
In our example, multiplying 5 by each term in the conjugate results in: \[ 5 \times (\sqrt{2} + 1) = 5\sqrt{2} + 5. \]
We then combine the results from the simplified numerator and denominators to get the final expression: \[ \frac{5\sqrt{2} + 5}{1} = 5\sqrt{2} + 5. \]Following these steps ensures our expression is fully simplified and rationalized.