Problem 26 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{\sqrt{20 y}}{y \sqrt{5}+1}\)

Step-by-Step Solution

Answer

The rationalized form is \(\frac{10y^{1.5} - \sqrt{20y}}{5y^2 - 1}\).

Step 1: Identify the Conjugate

To rationalize the denominator, we identify the conjugate of the denominator. The conjugate of the expression \(y \sqrt{5} + 1\) is \(y \sqrt{5} - 1\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \(y \sqrt{5} - 1\) to eliminate the square root from the denominator.\[\frac{\sqrt{20y}}{y \sqrt{5} + 1} \times \frac{y \sqrt{5} - 1}{y \sqrt{5} - 1} = \frac{\sqrt{20y}(y \sqrt{5} - 1)}{(y \sqrt{5})^2 - 1^2}\]

Step 3: Simplify the Denominator

Perform the difference of squares in the denominator: \((a+b)(a-b) = a^2 - b^2\).\[(y \sqrt{5})^2 - 1^2 = 5y^2 - 1\]

Step 4: Expand the Numerator

Distribute \(\sqrt{20y}\) in the numerator:\[\sqrt{20y} \times y \sqrt{5} - \sqrt{20y} \times 1 = y \sqrt{100y} - \sqrt{20y}\]Simplify further:\[y \cdot 10\sqrt{y} - \sqrt{20y} = 10y^{1.5} - \sqrt{20y}\]

Step 5: Write the Final Expression

Combine the simplified forms of the numerator and denominator:\[\frac{10y^{1.5} - \sqrt{20y}}{5y^2 - 1}\]This is the rationalized form of the given expression.

Key Concepts

Conjugate

When dealing with expressions that have square roots in the denominator, one effective method to eliminate them is using the conjugate. The conjugate of a binomial expression is formed by changing the sign between its two terms. For example, if you have the term \(a + b\), its conjugate would be \(a - b\). This simple switch is crucial because when we multiply a binomial by its conjugate, the resulting expression eliminates the square root terms, simplifying the expression significantly.

For the given expression \(y \sqrt{5} + 1\), the conjugate is \(y \sqrt{5} - 1\). By multiplying both the numerator and denominator of the fraction \(\frac{\sqrt{20y}}{y \sqrt{5} + 1}\) by this conjugate, we can simplify away the irrational numbers in the denominator."},{