Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ (-2 x)^{-2}(3 y)^{-3}(4 z)^{-2} $$

Step-by-Step Solution

Answer

\(\frac{1}{3456x^2 y^3 z^2}\)

Step 1: Convert negative exponents to positive exponents

To convert the negative exponents to positive exponents, we will use the property mentioned in the analysis. We have: \[ (-2 x)^{-2}(3 y)^{-3}(4 z)^{-2} = \frac{1}{(-2x)^{2}}\cdot\frac{1}{(3y)^{3}}\cdot\frac{1}{(4z)^{2}} \]

Step 2: Carry out the exponent calculations

Now we have positive exponents, let's carry out the calculations for each term: \[ \frac{1}{(-2x)^{2}}\cdot\frac{1}{(3y)^{3}}\cdot\frac{1}{(4z)^{2}} = \frac{1}{4x^2}\cdot\frac{1}{27y^3}\cdot\frac{1}{16z^2} \]

Step 3: Combine the fractions

Combine the three fractions into a single fraction: \[ \frac{1}{4x^2}\cdot\frac{1}{27y^3}\cdot\frac{1}{16z^2} = \frac{1}{4\cdot27\cdot16} \cdot \frac{1}{x^2\cdot y^3\cdot z^2} \]

Step 4: Simplify the constants

Multiply the constants in the numerator and the denominator: \[ \frac{1}{4\cdot27\cdot16} \cdot \frac{1}{x^2\cdot y^3\cdot z^2} = \frac{1}{3456} \cdot \frac{1}{x^2\cdot y^3\cdot z^2} \]

Step 5: Write the final expression

Now, we can write the final expression in the simplest form with positive exponents only: \[ \frac{1}{3456x^2 y^3 z^2} \] So, the simplified expression is: \(\frac{1}{3456x^2 y^3 z^2}\).

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they are quite straightforward once you understand a key rule. This rule states that a term with a negative exponent is the same as one divided by the term with the positive exponent. In other words,

• \(a^{-n} = \frac{1}{a^n}\)

This simple flip of perspective transforms complex equations into solvable ones. In our problem, each term with a negative exponent needed conversion. Here's the breakdown:

• \((-2x)^{-2} = \frac{1}{(-2x)^2}\)
• \((3y)^{-3} = \frac{1}{(3y)^3}\)
• \((4z)^{-2} = \frac{1}{(4z)^2}\)

By applying this rule, we quickly turned each negative exponent into a positive one, thus simplifying our task.

Fraction Simplification

Simplifying fractions is an essential skill in algebra. It allows you to condense complex fractions into more manageable forms. In our exercise, after converting the negative exponents, we ended up with multiple fractions:

• \(\frac{1}{4x^2}\)
• \(\frac{1}{27y^3}\)
• \(\frac{1}{16z^2}\)

The next step was to combine these into a single fraction. When multiplying fractions, multiply the numerators together and the denominators together. In this case, since each fraction had a numerator of 1, we simply multiplied the denominators.

• \(4 \cdot 27 \cdot 16\)
• \(x^2 \cdot y^3 \cdot z^2\)

Thus, we arrived at one fraction: \[\frac{1}{3456x^2 y^3 z^2}\]

Algebraic Expression Simplification

Simplifying algebraic expressions is like untangling a knot. The goal is to arrive at the simplest and cleanest version possible. In this exercise, after converting to positive exponents and combining fractions, we focused on the simplification of constants and variables.First, we performed the arithmetic on the constants in the denominator: 4 times 27 times 16, which calculated to 3456. Then, we gathered the remaining parts under a single fraction. Here's how:

• Combine constant numbers: \(4 \times 27 \times 16 = 3456\)
• Combine variables: spread the powers properly, \(x^2\), \(y^3\), and \(z^2\) staying as they are since they do not interact with each other in multiplication.

Thus, by neatly calculating and organizing, we achieved our final, simple result: \[\frac{1}{3456x^2 y^3 z^2}\]This expression correctly represents the original equation in its most simplified form using positive exponents only.