Problem 17 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{2 x}{5 y}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{10xy}}{5y}\)

Step 1: Rewrite the Expression

The original expression is \( \sqrt{\frac{2x}{5y}} \). To simplify this, we first rewrite it as \( \frac{\sqrt{2x}}{\sqrt{5y}} \) by using the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

Step 2: Identify the Conjugate

The goal is to eliminate the square root in the denominator. For this, we identify the "conjugate" of \( \sqrt{5y} \), which is itself, because \( \sqrt{5y} \times \sqrt{5y} = 5y \).

Step 3: Multiply Numerator and Denominator by Conjugate

Multiply both the numerator and the denominator by \( \sqrt{5y} \) to eliminate the square root in the denominator: \( \frac{\sqrt{2x} \times \sqrt{5y}}{\sqrt{5y} \times \sqrt{5y}} \).

Step 4: Simplify the Expression

Calculate the result: \( \sqrt{2x} \times \sqrt{5y} = \sqrt{10xy} \) for the numerator, and \( \sqrt{5y} \times \sqrt{5y} = 5y \) for the denominator. The expression simplifies to \( \frac{\sqrt{10xy}}{5y} \).

Key Concepts

Square Roots

Square roots are essential in mathematics, especially when dealing with radical expressions. A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. In mathematical terms, we represent square roots using the radical sign \( \sqrt{} \). This sign helps us identify roots, especially square roots, conveniently making calculations and simplifications possible.

Here are some important properties of square roots to remember:

• \( \sqrt{a^2} = a \) when \( a \geq 0 \)
• \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \)
• \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) (assuming \( b eq 0 \))

In the original exercise, we used \( \sqrt{\frac{2x}{5y}} = \frac{\sqrt{2x}}{\sqrt{5y}} \), allowing us to separate the numerator and denominator for the purposes of rationalization. Understanding these elementary properties aids greatly in simplifying complex expressions.

Simplifying Expressions

Simplifying expressions involves rewriting them in a form that is more straightforward or easier to work with. This process often involves combining like terms, reducing fractions, or eliminating radicals from the denominator. It’s a crucial skill for making algebraic problems less cumbersome.

In this exercise, simplifying began with rewriting the square root of a fraction as a fraction of square roots. This helped us recognize the need for rationalization by isolating the denominator part of the expression. By multiplying both the numerator and denominator by the conjugate of the denominator, we aim to remove any radicals in the denominator. This step is vital because having a rational number (without square roots) in the denominator makes it more acceptable according to mathematical convention and easier for further calculations. The result is a cleaner expression \( \frac{\sqrt{10xy}}{5y} \).

Effective simplification allows mathematicians and students alike to focus on essential elements of a problem without being bogged down by unnecessarily complex expressions.

Conjugates

Conjugates are pairs of expressions like\( a + b\sqrt{n} \) and\( a - b\sqrt{n} \) that, when multiplied, result in an integer or rational number by eliminating the square root term. This concept is used to rationalize denominators, an essential part of algebraic simplification.

The logic behind conjugates is that the product of two conjugate binomials is a difference of squares: \[(a + b\sqrt{n})(a - b\sqrt{n}) = a^2 - (b\sqrt{n})^2 = a^2 - b^2n\]In the exercise provided, the expression \( \sqrt{5y} \) is its own conjugate because when it’s multiplied by itself, we get: \[ \sqrt{5y} \times \sqrt{5y} = 5y \]This is precisely why we multiplied the numerator and the denominator by \( \sqrt{5y} \) to achieve a rational denominator of \( 5y \). Knowing how to use conjugates effectively is invaluable in both simplifying expressions and understanding the structure and properties of algebraic equations.