Problem 73 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \left(3 x^{2} y^{-3}\right)^{-2} $$

Step-by-Step Solution

Answer

\( \frac{y^6}{9x^4} \)

Step 1: Apply the Power of a Product Rule

The expression is \( \left(3 x^{2} y^{-3}\right)^{-2} \). Using the power of a product rule, \( (ab)^{m} = a^{m}b^{m} \), apply the exponent \(-2\) to each term inside the parentheses: \[ 3^{-2} \cdot (x^2)^{-2} \cdot (y^{-3})^{-2} \].

Step 2: Simplify Each Term Using Power of a Power Rule

Use the power of a power rule, \( (a^m)^n = a^{m \cdot n} \), to simplify each component:- \((x^2)^{-2} = x^{2 \times -2} = x^{-4}\)- \((y^{-3})^{-2} = y^{-3 \times -2} = y^{6}\).This gives the expression: \[ 3^{-2} \cdot x^{-4} \cdot y^{6} \].

Step 3: Simplify Each Term to Positive Exponents

Convert negative exponents to positive by using the rule \(a^{-m} = \frac{1}{a^m}\). Therefore, - \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)- \(x^{-4} = \frac{1}{x^4}\).This converts the expression to: \[ \frac{1}{9} \cdot \frac{1}{x^4} \cdot y^6. \]

Step 4: Combine Simplified Terms

Combine the simplified terms into a single fraction: \[ \frac{y^6}{9x^4}. \] This is achieved by multiplying the numerators together (\(y^6\)) and the denominators together (\(9x^4\)).

Key Concepts

Power of a Product

When you see an expression like \( (3x^2y^{-3})^{-2} \), you're dealing with an exponent applied to multiple terms inside the parentheses. This is where the power of a product rule comes in handy.
Power of a Product Rule Explained
The rule states: \((ab)^m = a^m b^m\). This means you will apply the exponent outside the parentheses to each term inside.

• For our example, each term inside the parentheses is: 3, \(x^2\), and \(y^{-3}\).
• Each term will be raised to the power of \(-2\).
• It becomes: \(3^{-2}\), \((x^2)^{-2}\), and \((y^{-3})^{-2}\).

It's like distributing the exponent to everything inside, making complex expressions simpler by breaking them down at the start.

Negative Exponents

Negative exponents can be a bit tricky at first, but they simplify your work when handled correctly. If you see a negative exponent, like in \((x^2)^{-2}\), it's signaling a reciprocal process.
Understanding Negative Exponents
The rule for negative exponents is: \(a^{-m} = \frac{1}{a^m}\). This means a negative exponent moves the base to the opposing side of a fraction line.

• For example, \(3^{-2} = \frac{1}{3^2}\).
• Similarly, \(x^{-4} = \frac{1}{x^4}\).

This effectively flips the term upside down in a fraction, turning negative exponents into positive ones and aiding overall simplification.

Simplification of Expressions

Simplifying expressions with exponents involves combining the elements we discussed to achieve a single, neat form. We take it step by step to ensure clarity and correctness.
Combining Simplified Terms
After applying the power of a product and addressing negative exponents, you're usually left with individual terms like \(\frac{1}{9}\), \(\frac{1}{x^4}\), and \(y^6\).
Next, you combine these into one clean expression:

• The numerators are multiplied: \(y^6\).
• The denominators are also multiplied together: \(9x^4\).

Putting these together, the expression becomes \(\frac{y^6}{9x^4}\). This approach ensures each part of your expression is neatly organized and fully simplified, facilitating clearer insights and ease of further analyses.