Problem 70 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt{9 x y}}{\sqrt{3 x^{2} y}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt{3x}}{x} \).

Step 1: Simplify the Radicals

First, simplify the expressions under the square roots. The numerator is \( \sqrt{9xy} \), and the denominator is \( \sqrt{3x^2y} \). These respectively simplify to \( \sqrt{9} \times \sqrt{x} \times \sqrt{y} \) and \( \sqrt{3} \times \sqrt{x^2} \times \sqrt{y} \). The individual square roots can be simplified as: \( \sqrt{9} = 3 \), \( \sqrt{x} = x^{1/2} \), \( \sqrt{y} = y^{1/2} \), \( \sqrt{x^2} = x \). This gives us:\[ 3\sqrt{xy} \text{ in the numerator and } \sqrt{3}xy^{1/2} \text{ in the denominator.} \]

Step 2: Rewrite the Fraction

Rewrite the simplified expression from Step 1 in the form: \[ \frac{3\sqrt{x} \sqrt{y}}{x\sqrt{3} \sqrt{y}} \] which can be simplified further by canceling out common terms. The common factor \( \sqrt{y} \) in the numerator and the denominator can be canceled out, resulting in \[ \frac{3\sqrt{x}}{x\sqrt{3}}. \]

Step 3: Rationalize the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{3} \) to get rid of the radical in the denominator: \[ \frac{3\sqrt{x}\sqrt{3}}{x\sqrt{3}\sqrt{3}} = \frac{3\sqrt{3x}}{3x}. \] Since \( \sqrt{3}\sqrt{3} = 3 \), the denominator becomes \( 3x \). The fraction simplifies as: \[ \frac{\sqrt{3x}}{x}. \]

Step 4: Simplify the Expression Further

The fraction \( \frac{\sqrt{3x}}{x} \) can be rewritten as: \[ \frac{1}{x} \times \sqrt{3x} = \frac{\sqrt{3x}}{x}. \] No more simplifications can be done, as the fraction is already in its simplest form.

Key Concepts

Rationalization

Rationalization is a crucial algebraic technique used to eliminate square roots or other radicals from the denominator of a fraction. The main goal is to simplify the expression to make it easier to work with, especially when performing arithmetic operations. In this exercise, you encountered a fraction

• Step 1: Given a radical in the denominator such as \(\sqrt{3}x\), rationalize it by multiplying the numerator and the denominator by \(\sqrt{3}\) so that the denominator becomes a rational number \((\sqrt{3}\times\sqrt{3} = 3)\).
• Step 2: The process results in a new fraction with an easier form for calculations. This is because radicals in denominators can complicate addition and subtraction of fractions, making rationalization an essential simplification step.
• Step 3: Understanding this technique is vital since it's frequently applied in algebraic manipulation involved in calculus problems and solving equations.

A key concept to remember is that rationalization changes the form but not the value of the expression.

Square Roots

In mathematics, a square root is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\).Square roots can often appear under radicals in algebraic expressions. For example, in the numerator \(\sqrt{9xy}\) of this problem, you break down the expression as

• Find individual roots: \(\sqrt{9} = 3\), \(\sqrt{x}\), and \(\sqrt{y}\).
• Acknowledge that the square root function works by reversing the operation of squaring. This is why \(\sqrt{x^2} = x\).

Understanding square roots helps simplify expressions and solve equations by isolating variables. It's also essential for working through exercises involving radicals and ensuring correct manipulation during simplification steps.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are key components of algebraic problems. In this exercise, both the numerator and denominator contain radical expressions, initially written as \(\sqrt{9xy}\) and \(\sqrt{3x^2y}\).To simplify radical expressions, break them down into their basic components:

• Observe that \(\sqrt{9xy} = 3\sqrt{xy}\), demonstrating how square roots pull perfect squares like 9 out of the radical.
• Simplifying the denominator \(\sqrt{3x^2y}\) is similar: \(\sqrt{x^2} = x\), so it becomes \(x\sqrt{3y}\).

Radical expressions are simplified by factoring out and simplifying any perfect squares or common factors, which sometimes involves rationalizing to remove radicals from the denominator. This method is fundamental in algebra for clarifying expressions and ensuring all operations follow a logical, systematic approach.