Problem 13 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\frac{\sqrt{12 x^{4} y}}{\sqrt{3 x^{2} y^{5}}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2x}{y^2} \).

Step 1: Rewrite the Radicals

Start by breaking down the radicals in the numerator and denominator. We have:\[ \frac{\sqrt{12 x^4 y}}{\sqrt{3 x^2 y^5}} = \frac{\sqrt{12} \cdot \sqrt{x^4} \cdot \sqrt{y}}{\sqrt{3} \cdot \sqrt{x^2} \cdot \sqrt{y^5}}. \]

Step 2: Simplify Each Component

Simplify each square root separately. First, simplify \(\sqrt{12}\) and \(\sqrt{3}\):\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \] and \[ \sqrt{3} = \sqrt{3}. \] Now simplify \(\sqrt{x^4}\) and \(\sqrt{x^2}\):\[ \sqrt{x^4} = x^2 \] and \[ \sqrt{x^2} = x. \] For \(\sqrt{y}\) and \(\sqrt{y^5}\), write them as powers of \(y\):\[ \sqrt{y} = y^{1/2} \] and \[ \sqrt{y^5} = y^{5/2}. \]

Step 3: Rewrite the Expression

Substitute the simplified components back into the expression:\[ \frac{2\sqrt{3} \cdot x^2 \cdot y^{1/2}}{\sqrt{3} \cdot x \cdot y^{5/2}}. \]

Step 4: Cancel Common Factors

Cancel out any common factors from the numerator and the denominator. The common factor here is \(\sqrt{3}\), leaving us with:\[ \frac{2x^2y^{1/2}}{xy^{5/2}}. \] Simplify further by canceling \(x\) and using the property of exponents:\[ 2x^{2-1}y^{1/2-5/2} = 2x^{1}y^{-2} = 2x\cdot \frac{1}{y^2}. \]

Step 5: Simplify Further if Possible

Ensure the expression is simplified further so no negative exponents remain: \[ \frac{2x}{y^2}. \] This is the expression with a rationalized denominator.

Key Concepts

Simplification of Expressions

In mathematics, simplifying expressions is a crucial skill. It involves rewriting an expression in its simplest form. This process usually makes the expression more comprehensible and easier to work with.
For the exercise at hand, we began with a fraction involving square roots, which needed simplification.

• Separate Elements: First, break down complex radicals into simpler parts. This might involve identifying perfect squares within a square root.
• Combine Like Terms: Once simplified, the component parts can be recombined, keeping in mind to reduce where possible. Cancellation occurs if both the numerator and denominator share similar terms.

The primary goal here is to present the expression in its simplest equivalent form, so it's more manageable.

Properties of Exponents

The properties of exponents are like a toolbox for manipulating expressions involving powers. They help us simplify and evaluate radicals effectively.

• Product Rule: This property states that when you multiply two powers with the same base, you add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient Rule: When dividing powers with the same base, you subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Negative Exponent Rule: It indicates a reciprocal relationship: \(a^{-n} = \frac{1}{a^n}\).

Applying these rules, we simplified \(x^{2-1}\) to \(x\) and transformed \(y^{1/2 - 5/2}\) to \(y^{-2}\). Rewriting negative exponents as positive ones is important for further simplification.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. Simplifying radical expressions often means rationalizing them, removing any roots from the denominator.

• Breaking Down Radicals: We started by extracting square roots of numbers, like \(\sqrt{12}\) being expressed as \(2\sqrt{3}\).
• Rationalizing the Denominator: This process removes radicals from a fraction's denominator. Multiply both the numerator and denominator by the necessary radical value to achieve this.

In our exercise, rationalizing was while ensuring no radicals remained in the denominator. Squeezing through these steps demands precision and clarity to make the expression easier to work with.