Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \frac{2^{2} u^{-2}\left(v^{-1}\right)^{3}}{3^{2}\left(u^{-3} v\right)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression, written with positive exponents only, is \(\frac{4}{9u^{8}v^{5}}\).

Step 1: Rewrite the expression with positive exponents

To rewrite the expression using positive exponents only, we need to recall that a negative exponent indicates-inverse (i.e., a^(-n) = 1/a^n). In our case, we see that we have u^(-2) and v^(-1). We can rewrite them using positive exponents: $$ \frac{2^{2}\cdot \frac{1}{u^{2}} \cdot \left(\frac{1}{v}\right)^{3}}{3^{2}\cdot\left(\frac{1}{u^{3}} \cdot v\right)^{2}} $$

Step 2: Simplify the expression

Now, we simplify the powers in the expression: $$ \frac{4\cdot \frac{1}{u^{2}} \cdot \frac{1}{v^{3}}}{9\cdot\left(\frac{1}{u^{6}} \cdot v^{2}\right)} $$

Step 3: Rewrite the expression as a single fraction

Next, we rewrite the expression as a single fraction by multiplying the numerators and the denominators respectively. $$ \frac{4 \cdot 1 \cdot 1 }{u^{2} \cdot v^{3} \cdot 9 \cdot u^{6} \cdot v^{2}} $$

Step 4: Combine like terms

Now, we combine like terms by adding their exponents. $$ \frac{4}{9 \cdot u^{2+6} \cdot v^{3+2}} $$

Step 5: Final simplification

Finally, we simplify the expression by combining the exponents of similar variables. $$ \frac{4}{9u^{8}v^{5}} $$ The simplified expression, written with positive exponents only, is \(\frac{4}{9u^{8}v^{5}}\).

Key Concepts

Positive Exponents

Understanding positive exponents is crucial when simplifying algebraic expressions. Exponents are shorthand for repeated multiplication. A positive exponent, such as \(2^3\), tells you how many times to multiply the base (2) by itself: \(2 \times 2 \times 2 = 8\). But what happens with negative exponents?

Negative exponents, like \(a^{-n}\), indicate the reciprocal of the base raised to a positive exponent. For instance, \(a^{-n} = \frac{1}{a^n}\). Therefore, when simplifying an expression with negative exponents, you must first rewrite them as fractions with positive exponents to make the rest of the simplification process more straightforward. Remember that \(a^0 = 1\), regardless of the value of \(a\), except when \(a\) is zero.

Rewriting expressions with negative exponents helps you to see the structure of the expression more clearly and sets you up for the further steps of combining like terms and final simplification to achieve an answer with only positive exponents.

Rewriting Expressions

Rewriting expressions is an essential step in the simplification process. It involves rearranging and transforming expressions into a more manageable form without changing its value. This process allows you to clarify complex expressions and prepares you for further operations such as combining like terms or simplifying fractions.

For example, rewriting \(\frac{a^{-n}}{b^{-m}}\) as \(\frac{b^m}{a^n}\) using positive exponents can make it easier to understand and manipulate. The goal of rewriting is to present the algebraic expression in its simplest and most comprehensible form. Remember to apply the order of operations (PEMDAS/BODMAS) correctly when rewriting expressions. Sometimes you may need to expand or factor expressions; other times, you might need to distribute terms or consolidate fractions. Each step contributes to making the overall process of simplification smoother and more intuitive.

Combining Like Terms

Combining like terms is a technique used to simplify algebraic expressions or equations. Like terms are terms that have the same variables raised to the same powers, even if their coefficients are different. When you combine like terms, you add or subtract the coefficients, while the variable part remains unchanged.

For instance, with the terms \(2x^2\) and \(3x^2\), you can combine them to get \(5x^2\) because they have the same variable \(x\) raised to the same power. In contrast, \(2x^2\) and \(3x\) are not like terms due to the different exponents on \(x\), and thus cannot be combined in this manner.

When simplifying an expression, after rewriting all terms with positive exponents, look for like terms to combine. This process will often lead you to a more concise form of the expression and bring you one step closer to your final simplified answer. It's like gathering apples together and oranges together; you wouldn't mix the two. Always ensure that you are only combining terms that have exactly matching variable parts.