Problem 63 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt{9 x^{-4} y^{6}} $$

Step-by-Step Solution

Answer

\(\frac{3y^3}{x^2}\)

Step 1: Simplify the Radicand

The expression under the square root is \(9x^{-4}y^6\). Start by simplifying this expression. Notice that \(9\) is a perfect square since \(9 = 3^2\). The expression becomes \((3^2)x^{-4}y^6\).

Step 2: Apply Square Root to Each Component

Apply the square root to each component under the radical: \(\sqrt{(3^2)x^{-4}y^6} = \sqrt{3^2} \times \sqrt{x^{-4}} \times \sqrt{y^6}\).

Step 3: Evaluate Square Roots

Calculate each square root separately: \(\sqrt{3^2} = 3\), \(\sqrt{x^{-4}} = x^{-2}\), and \(\sqrt{y^6} = y^3\). This results in \(3x^{-2}y^3\).

Step 4: Rationalize if Necessary

Check if the expression requires rationalization. In this case, the expression \(3x^{-2}y^3\) doesn't have a denominator to rationalize. However, it can be rewritten as \(\frac{3y^3}{x^2}\) to eliminate the negative exponent.

Key Concepts

Square Roots

Understanding square roots is fundamental to simplifying expressions like \(\sqrt{9x^{-4}y^6}\). A square root essentially asks, "what number, when multiplied by itself, gives this number?" For example, the square root of 9 is 3, because \(3 \times 3 = 9\).

When simplifying expressions under a square root, look for perfect squares — numbers like 4, 9, 16, 25, etc. In our expression, the number 9 is a perfect square. Beyond numbers, when dealing with variables, express them in their simplest square form, if possible.

• \(x^{-4}\) can be seen as \((x^{-2})^2\)
• \(y^6\) can be viewed as \((y^3)^2\)

When applying square roots to these components, you're cutting their exponents in half (e.g., square root of \((y^3)^2\) is \(y^3\)). Keep this in mind to simplify correctly.

Rationalizing the Denominator

Rationalizing the denominator means rewriting a fraction so that there are no square roots (or other radicals) in the denominator. Typically, this doesn't apply unless you have a fraction with a square root in its bottom part. However, in expressions involving negative exponents, rationalization becomes part of clarifying the form.

In our exercise, we initially face \(3x^{-2}y^3\). The negative exponent hints at a form of division, expressed as \(\frac{3y^3}{x^2}\). This step rationalizes or simplifies away the negative exponent, preventing potential complexity in interpretation or calculation later in math expressions or solving further equations.

Negative Exponents

A negative exponent indicates a reciprocal. It turns positive when you move it from the numerator to the denominator, or vice versa. For instance, \(x^{-2}\) is equivalent to \(\frac{1}{x^2}\).

In the context of our exercise, \(x^{-4}\) converts to \(\frac{1}{x^4}\) before simplifying reduces it to \(x^{-2}\), later written in a rational form as \(\frac{1}{x^2}\). Recognizing negative exponents is crucial to adjusting expressions into clearer formats and is frequently paired with rationalization in problem solving.

• Move the base with the negative exponent "across the fraction line" to make it positive.
• Provide clarity and simplification as these negative powers can often cause errors if left unchecked in expressions.