Problem 67 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt{10 y^{2}}}{\sqrt{2 y^{3}}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt{5y}}{y} \).

Step 1: Simplify the Expression

Start by simplifying the expression under the radicals. The given expression is \( \frac{\sqrt{10 y^2}}{\sqrt{2 y^3}} \). This can be simplified by dividing the radicands: \( \sqrt{\frac{10 y^2}{2 y^3}} \).

Step 2: Simplify Under the Radical

Divide inside the square root to simplify further. This becomes \( \sqrt{\frac{10}{2} \cdot \frac{y^2}{y^3}} = \sqrt{5 \cdot \frac{1}{y}} = \sqrt{\frac{5}{y}} \).

Step 3: Rationalize the Denominator

To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{y} \) to eliminate the square root in the denominator. This gives us \( \frac{\sqrt{5}}{\sqrt{y}} \times \frac{\sqrt{y}}{\sqrt{y}} = \frac{\sqrt{5y}}{y} \).

Step 4: Final Simplified Form

The expression \( \frac{\sqrt{5y}}{y} \) is now fully simplified with a rationalized denominator, since there are no square roots in the denominator.

Key Concepts

Simplifying Radicals

Radicals are expressions containing square roots, cube roots, or higher roots. When simplifying radicals, the goal is to make them as clean and concise as possible. In the context of our problem, we are dealing with square roots.
Steps to Simplify Radicals

• Combine like terms inside the radicals.
• For square roots, check if the number or terms inside can be broken into squares. For example, \( \sqrt{12} \) can be split into \( \sqrt{4 \cdot 3} \), which simplifies to \( 2\sqrt{3} \).
• Always look for opportunities to reduce fractions under the root before attempting to simplify further. This can make future steps straightforward.

Simplifying the radicals helps in making the subsequent steps, like rationalizing the denominator, more manageable. It also ensures that the expression maintains its mathematical equivalence in a simpler form.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For instance, \( \sqrt{4} \) equals 2 because \( 2 \times 2 = 4 \).
Remember These Tips:

• Square roots can be both simplified and rationalized, but start with simplification first.
• If necessary, separate variables and constants under the square root to simplify separately.
• Always aim to express the square root in its most basic form by removing any perfect squares.

Square roots often appear in various calculations involving geometry, algebra, and even in real-world applications. Handling them correctly is essential, especially when preparing to rationalize denominators or simplify complex expressions.

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator, the denominator, or both. If the denominator contains a square root, it can be converted into a simpler form through rationalization.
Key Ideas for Rationalizing:

• Multiply the numerator and denominator by a radical that will eliminate the square root in the denominator. This is often the same radical as in the denominator.
• Continue the process until the denominator is free of roots and is a rational number.
• Ensure the resulting expression is still equivalent to the original.

For instance, to rationalize \(\frac{\sqrt{5y}}{y}\), multiply both the numerator and the denominator by \(\sqrt{y}\), resulting in \(\frac{\sqrt{5y}}{y}\) to \(\frac{\sqrt{5y} \cdot \sqrt{y}}{y \cdot \sqrt{y}}\). This simplifies to \(\frac{\sqrt{5y^2}}{y}\), which is an expression where the denominator has been rationalized.