Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(\frac{x^{6}}{9 y^{-4}}\right)^{-1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3y^2}{x^3} \).

Step 1: Understand the Problem

We need to simplify the given expression \( \left(\frac{x^{6}}{9 y^{-4}}\right)^{-1 / 2} \). This involves rewriting the expression without negative exponents and simplifying it as much as possible.

Step 2: Apply the Negative Exponent Rule

The expression has a negative exponent, \(-1/2\). Recall that \(a^{-n} = \frac{1}{a^n}\). Therefore, \[\left(\frac{x^{6}}{9 y^{-4}}\right)^{-1 / 2} = \frac{1}{\left(\frac{x^{6}}{9 y^{-4}}\right)^{1/2}} = \left(\frac{9 y^{-4}}{x^{6}}\right)^{1/2}\]

Step 3: Simplify the Fraction Inside the Radical

Rewrite the fraction inside the square root:\[\frac{9}{x^6} \cdot y^4\] where we used \(y^{-4} = \frac{1}{y^4}\) and then simplified it by considering it on the numerator.

Step 4: Apply the Square Root to Each Part

The expression is now a square root, which can be applied to each part separately:\[\sqrt{\frac{9y^4}{x^6}} = \frac{\sqrt{9} \cdot \sqrt{y^4}}{\sqrt{x^6}}\] Simplifying gives us \(\frac{3y^2}{x^3}\) because \(\sqrt{9} = 3\), \(\sqrt{y^4} = y^2\), and \(\sqrt{x^6} = x^3\).

Step 5: Final Simplification

Combine the simplified parts from the previous step, and we are left with the expression:\[\frac{3y^2}{x^3}\] This is the simplified form of the original expression.

Key Concepts

Simplifying Expressions

When you encounter expressions with exponents and radicals, the goal is often to simplify them, making them easier to work with. Simplifying expressions is a process where you transform a complex expression into a more manageable form without changing its value.

In the problem \( \left(\frac{x^{6}}{9 y^{-4}}\right)^{-1 / 2} \), we started by recognizing that the expression involves fractions, exponents, and radicals. The first step in simplification is to eliminate any negative exponents and apply rules like the power of a power or power of a product rule.

After addressing negative exponents, it's crucial to manage any radicals. Each part of the fraction under a radical can be simplified separately. This is done by breaking down the expression into its simplest form, using square roots for any squared terms.

Negative Exponents

Understanding negative exponents can simplify expressions significantly. A negative exponent indicates that the base should be taken as the reciprocal with a positive exponent:

• For example, \( a^{-n} = \frac{1}{a^n} \).

In the exercise \( \left(\frac{x^{6}}{9 y^{-4}}\right)^{-1 / 2} \), we apply this rule to handle the negative \( -1/2 \) exponent.

By doing so, the expression is rewritten as a reciprocal: \( \frac{1}{\left(\frac{x^{6}}{9 y^{-4}}\right)^{1/2}} = \left(\frac{9 y^{-4}}{x^{6}}\right)^{1/2} \). In the radical expression, remove negative exponents by placing them on the opposite side of the fraction, making calculations easier in successive steps.

Handling negative exponents correctly thus simplifies expressions and can reduce computational errors.

Square Roots

Square roots are a type of radical that you will encounter often. They simplify when each term within the radical is broken down into its factors. In our problem, we attained a form where the expression became \( \sqrt{\frac{9y^4}{x^6}} \).

• To simplify this, break it down: \( \sqrt{9} \), \( \sqrt{y^4} \), and \( \sqrt{x^6} \).
• The square root of \( 9 \) is \( 3 \), \( \sqrt{y^4} \) results in \( y^2 \), and \( \sqrt{x^6} \) simplifies to \( x^3 \).

When simplifying radicals, it's useful to factor terms into perfect squares.

Perfect squares make taking square roots straightforward. The final expression \( \frac{3y^2}{x^3} \) shows the result of these simplifications. Comprehending how to simplify square roots helps in demystifying complex expressions and solving them effectively.