Problem 55 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(8rs^4\).

Step 1: Understand the Base Expression

The expression given is \((rs)^{-2}(2rs^2)^3\). Our goal is to simplify this expression and ensure there are no negative exponents in the result.

Step 2: Apply the Power Distribution

Distribute the powers across the terms in each part of the expression. For the second term, \((2rs^2)^3\), apply the power of 3 to each factor inside the parentheses:- The coefficient becomes \(2^3\).- The term \(r\) becomes \(r^3\).- The term \(s^2\) becomes \(s^{6}\). So, \((2rs^2)^3 = 8r^3s^6\).

Step 3: Combine and Simplify

Combine the expressions by multiplying them, \((rs)^{-2} \cdot 8r^3s^6 = 8(r)^{-2} (s)^{-2} (r)^{3} (s)^{6} \). Combine the exponents for like terms (\(r\) and \(s\)):- For \(r\): \(r^{-2} \cdot r^3 = r^{(-2+3)} = r^1 = r\).- For \(s\): \(s^{-2} \cdot s^6 = s^{(-2+6)} = s^4\).

Step 4: Final Simplified Expression

Finally, multiply the coefficient with the combined terms:\(8 \cdot r \cdot s^4 = 8rs^4\).This expression has no negative exponents and is fully simplified.

Key Concepts

Negative Exponents

Negative exponents can initially seem confusing, but they simplify into a fraction. Essentially, when you have a negative exponent, like in \((rs)^{-2}\), it translates into the reciprocal of the base raised to the positive exponent. This transforms the expression into \(\frac{1}{(rs)^2}\). The negative sign dictates the reciprocal action.

• Rule: For any non-zero value \(a\), \(a^{-n} = \frac{1}{a^n}\).

Negative exponents help us move terms between numerator and denominator, thereby allowing the simplification of expressions that might otherwise seem complicated.

Power Distribution

Power distribution refers to the process of applying an exponent to all components within a parenthesis. This concept is crucial in simplifying expressions such as \((2rs^2)^3\). Each factor inside the parenthesis receives the power of 3 individually:

• When applying the power of 3, \(2^3 = 8\).
• The variable \(r\) becomes \(r^3\).
• The variable \(s^2\) becomes \(s^{6}\).

This results in \((2rs^2)^3 = 8r^3s^6\), expanding the expression and simplifying subsequent steps.Understanding power distribution helps maintain clarity in longer expressions, ensuring each factor is considered properly.

Combining Like Terms

Combining like terms is a method of simplifying expressions by merging terms with the same variable raised to any power. Consider the expression: \((rs)^{-2} \cdot 8r^3s^6\). Here, like terms involve the variable \(r\) with exponents, and the variable \(s\) also with exponents. Here’s what happens:

• For \(r\): Combine \(r^{-2}\) with \(r^3\) to get \(r^{1}\) using the addition of exponents \(-2 + 3 = 1\).
• For \(s\): Combine \(s^{-2}\) with \(s^6\) to yield \(s^4\) using the sum of exponents \(-2 + 6 = 4\).

Simplification using combining like terms reduces redundancy and consolidates longer polynomial expressions into more manageable forms to work with, facilitating easier manipulation and further operations.

Exponent Rules

Exponent rules are essential guidelines that govern how exponents interact in mathematical expressions. These rules help navigate operations involving exponents efficiently.

• Product of Powers Rule: When multiplying terms with the same base, you add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power Rule: When raising an exponential expression to another power, multiply the exponents: \((a^m)^n = a^{mn}\).

These are just a couple of important exponent rules. Applying these rules effectively can transform complex expressions into simpler forms, readying them for further calculation or expression solutions.Mastering exponent rules allows for swift simplification and transformation of algebraic structures, ultimately leading to more efficient problem-solving capabilities.