Problem 66 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \frac{\left(r s^{2}\right)^{3}}{\left(r^{-3} s^{2}\right)^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(r^9 s^2\).

Step 1: Simplify the Numerator

The numerator of the expression is \((rs^2)^3\). To simplify this, distribute the exponent across each factor in the parentheses. This yields \(r^3\) and \((s^2)^3 = s^6\). Thus, the numerator simplifies to \(r^3 s^6\).

Step 2: Simplify the Denominator

The denominator is \((r^{-3} s^2)^2\). Distribute the exponent across each factor: \((r^{-3})^2\) becomes \(r^{-6}\) and \((s^2)^2\) becomes \(s^4\). The denominator simplifies to \(r^{-6} s^4\).

Step 3: Divide the Expressions

Divide the simplified numerator \(r^3 s^6\) by the simplified denominator \(r^{-6} s^4\). Use the property \(a^m / a^n = a^{m-n}\) for each variable. This gives: \(r^{3 - (-6)} = r^{3+6} = r^9\) and \(s^{6-4} = s^2\).

Step 4: Final Expression

Combine the results from the division: the entire expression simplifies to \(r^9 s^2\).

Key Concepts

Negative Exponents

Negative exponents can seem a bit confusing at first, but they're easier to understand when you think of them as a way of representing division or fractions in a different format. When you have a negative exponent, like in the example of the denominator \(r^{-3}\), it essentially means you are dealing with the reciprocal of the base raised to the positive version of the exponent. In simpler terms: \(r^{-3} = \frac{1}{r^3}\).

• Whenever you see a negative exponent, try thinking "how can I flip this?"
• Remembering this can help you switch between divisions and multiplications easily, making calculations quicker.
• This trick is fundamental because it helps with simplifying complex expressions in algebra.

Practicing manipulating negative exponents will make problems like these more approachable in the long run.

Exponent Rules

Exponent rules are powerful tools in algebra. Simplifying expressions with exponents relies heavily on understanding a few key rules:

• Product of Powers Rule: When multiplying like bases, you add the exponents together (e.g., \(a^m \times a^n = a^{m+n}\)).
• Power of a Power Rule: When raising a power to another power, multiply the exponents (e.g., \( (a^m)^n = a^{m\times n}\)).
• Quotient of Powers Rule: When dividing like bases, subtract the exponents (e.g., \( \frac{a^m}{a^n} = a^{m-n}\)).

In the exercise provided, these rules help break down both the numerator and the denominator before combining them to simplify the entire expression. By using these rules systematically, you can tackle complicated expressions step by step, making them much simpler.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions, but they involve expressions with variables. Solving them requires manipulating both the numerator and the denominator using algebraic principles such as factoring and applying exponent rules.In the exercise, we start by simplifying each part separately:- **For the numerator** \( (rs^2)^3 \), each term inside the parentheses is raised to the third power. This results in \(r^3 s^6\) using the power of a power rule.- **For the denominator** \( (r^{-3} s^2)^2 \), we apply the same rule but pay attention to the negative exponent in \(r^{-3}\), turning it into \(r^{-6}\) because of \( (a^m)^n = a^{m\times n}\).By simplifying both the numerator and the denominator, you can then use the quotient of powers rule to reduce the fraction further, finding the final simplified expression \(r^9 s^2\). Understanding these steps is essential when facing any algebraic fractions involving variables and exponents.