Problem 28 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

\( \dfrac{u^3 v^6}{27} \)

Step 1: Apply properties of exponents to each term inside the parentheses and separate the variables

By applying the power rule, we multiply the exponents of each term inside the parentheses by -3. \( \left( \left( 3 \right)^{-3} \left( u^{-1} \right)^{-3} \left( v^{-2} \right)^{-3} \right) \)

Step 2: Simplify the expression inside the parentheses

Simplify each term and write them separately. \( 3^{-3} \cdot u^{-1 \cdot (-3)} \cdot v^{-2 \cdot (-3)} = \dfrac{1}{3^3} \cdot u^3 \cdot v^6 \)

Step 3: Combine terms to get the final simplified expression

Finally, merge the terms inside the expression to get the simplified result. \( \dfrac{1 \cdot u^3 \cdot v^6}{3^3} = \dfrac{u^3 v^6}{27} \)

Key Concepts

Power Rule

The power rule is a key concept when dealing with exponents, especially in expressions. It allows us to simplify complex expressions by multiplying the exponents. When raising a power to another power, you multiply the exponents according to the power rule. For instance, if you have \((x^a)^b\), it becomes \(x^{a \cdot b}\).
The power rule makes simplifying expressions much more straightforward. In the given expression, \( (3u^{-1}v^{-2})^{-3} \), each term inside the brackets has its exponent multiplied by -3. Let's break down each term:

• \(3\) raised to power -3 becomes \(3^{-3}\)
• \(u^{-1}\) raised to power -3 becomes \(u^{-1 \cdot (-3)} = u^3\)
• \(v^{-2}\) raised to power -3 becomes \(v^{-2 \cdot (-3)} = v^6\)

Recognizing and applying the power rule simplifies the computation significantly.

Simplifying Expressions

Simplifying an expression often involves using the power rule as well as other fundamental properties of exponents. It helps express terms in the simplest form possible. For instance, in our exercise, after applying the power rule, we get \(3^{-3} \cdot u^3 \cdot v^6\). To simplify this expression further, focus on converting any negative exponents into positive exponents.
Negative exponents, such as \(3^{-3}\), can be rewritten as fractions: \(\frac{1}{3^3}\). This process entails placing the term with the negative exponent in the denominator to make it positive. So, \(3^{-3}\) becomes \(\frac{1}{27}\).

• Arrange the simplified terms: \(u^3\) and \(v^6\) remain in the numerator.
• Final expression: \(\frac{u^3 \cdot v^6}{27}\)

The simplification is complete when each term is as simple as possible, with no negative exponents at play.

Positive Exponents

Positive exponents are often preferred in final expressions. They provide a clearer and more standard representation of mathematical solutions. When simplifying expressions, converting all negative exponents to positive ones is essential.
Remember, a negative exponent suggests the inverse of that number. For example, \(a^{-b}\) can be rewritten as \(\frac{1}{a^b}\), leading to positive exponents. This method was applied to the original expression \(3^{-3}\) to convert it into \(\frac{1}{3^3} = \frac{1}{27}\).

• When simplifying, ensure all exponents are positive for clarity and standardization.
• The solution \(\frac{u^3 v^6}{27}\) represents fully positive exponents, maintaining a clean and universally understandable form.

Achieving a solution with positive exponents streamlines communication and understanding in math contexts.