Problem 25 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{\sqrt{5 x}}{\sqrt{5 x}-2}\)

Step-by-Step Solution

Answer

\( \frac{5x + 2\sqrt{5x}}{5x - 4} \) is the simplified form with a rationalized denominator.

Step 1: Identify the Expression to Rationalize

The given expression is \( \frac{\sqrt{5x}}{\sqrt{5x} - 2} \). The goal is to eliminate the square root in the denominator by rationalizing it.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( \sqrt{5x} - 2 \) is \( \sqrt{5x} + 2 \). So we have:\[\frac{\sqrt{5x}}{\sqrt{5x} - 2} \times \frac{\sqrt{5x} + 2}{\sqrt{5x} + 2} = \frac{\sqrt{5x}(\sqrt{5x} + 2)}{(\sqrt{5x} - 2)(\sqrt{5x} + 2)}\]

Step 3: Simplify the Denominator

Apply the difference of squares formula, \((a-b)(a+b) = a^2 - b^2\), to the denominator:\[(\sqrt{5x})^2 - (2)^2 = 5x - 4\]

Step 4: Simplify the Numerator

Distribute \(\sqrt{5x}\) across the terms in the numerator:\[\sqrt{5x} \cdot \sqrt{5x} + \sqrt{5x} \cdot 2 = 5x + 2\sqrt{5x}\]

Step 5: Write the Simplified Expression

Combine the simplified numerator and denominator from Step 3 and Step 4:\[\frac{5x + 2\sqrt{5x}}{5x - 4}\] This is the expression with a rationalized denominator.

Key Concepts

rationalize the denominator

Rationalizing the denominator is an essential skill in Algebra 2. It involves transforming a fraction so that the denominator is a rational number. In simple words, we want to get rid of any square roots or irrational numbers in the denominator.

To achieve this, we use a method involving the conjugate. The conjugate of a binomial expression like \(a - b\) is \(a + b\). This method is effective because multiplying a binomial by its conjugate results in the difference of squares, which eliminates the square roots. In our problem, we dealt with \(\sqrt{5x} - 2\), and its conjugate is \(\sqrt{5x} + 2\).

By multiplying both the numerator and the denominator by this conjugate, the denominator becomes a rational number.

difference of squares

The difference of squares is a handy algebraic identity that states \((a - b)(a + b) = a^2 - b^2\). This identity is crucial when rationalizing denominators because it allows us to remove square roots.

In the given problem, after multiplying by the conjugate, the denominator transforms as follows: \((\sqrt{5x})^2 - (2)^2 = 5x - 4\). Notice how the radicals disappear as a result of this operation. This is because \((\sqrt{5x})^2\) simplifies to \(5x\), and we subtract \(4\), the square of \(2\).

This simplification process of replacing irrational denominators with rational numbers is a powerful technique, and recognizing the pattern of the difference of squares can make solving these problems straightforward.

simplify expressions

Simplifying expressions involves reducing them to their simplest form without altering their value. After rationalizing the denominator, it's essential to ensure your expression is simplified.

In the numerator, we distribute \(\sqrt{5x}\) across the terms of the conjugate. This gives us \(\sqrt{5x} \times \sqrt{5x} + 2 \times \sqrt{5x} = 5x + 2\sqrt{5x}\). We transform each part of the numerator by performing basic distribution rules to make it clean and simple.

After both numerator and denominator have been simplified, the entire fraction can be written in its simplest form: \(\frac{5x + 2\sqrt{5x}}{5x - 4}\).

• This form is concise.
• It has clear rational components.
• It portrays the desired outcome of rationalization.

The goal is to have no further simplifications possible, making the expression clear and comprehensible.