Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ \left(\frac{27 x^{-3} y^{2}}{8 x^{-2} y^{-5}}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3y^2}{2x^{1/3}}\).

Step 1: Simplify the fraction inside the parentheses

To simplify the fraction, we will use the rule of exponents \(a^{-n} = \frac{1}{a^n}\): \[ \frac{27 x^{-3} y^{2}}{8 x^{-2} y^{-5}} = \frac{27 x^{-3} y^{2}}{8 \frac{1}{x^2} \frac{1}{y^5}} = \frac{27 x^{-3} y^{2} x^2 y^5}{8} \]

Step 2: Apply the power (1 / 3) to each term inside the parentheses

Now apply the power of 1/3 to each term, remembering that \((a^m)^n = a^{mn}\): \[ \left(\frac{27 x^{-3} y^{2} x^2 y^5}{8}\right)^{1/3} = \frac{27^{1/3} x^{-3(1/3)} y^{2(1/3)} x^{2(1/3)} y^{5(1/3)}}{8^{1/3}} \]

Step 3: Simplify the result

Simplify the expression by combining the like terms and calculating the values of exponents: \[ \frac{3 x^{-1} y^{2/3} x^{2/3} y^{5/3}}{2} = \frac{3 (x^{-1+2/3}) (y^{2/3 + 5/3})}{2} = \frac{3x^{-1/3}y^2}{2} \] Since we need to write the answer using positive exponents only, we use the rule of exponents again: \[ \frac{3x^{-1/3}y^2}{2} = \frac{3y^2}{2x^{1/3}} \] The final answer is: \[ \frac{3y^2}{2x^{1/3}} \]

Key Concepts

Simplifying Fractions

Simplifying fractions involves making them as simple as possible. In this exercise, fractions are connected to expressions with exponents. To simplify, you apply the rules of exponents. For example, an exponent like \(a^{-n}\) translates to \(\frac{1}{a^n}\). When simplifying a fraction like \(\frac{27x^{-3}y^2}{8x^{-2}y^{-5}}\), you first transform each term using the negative exponent rule.

• Convert negative powers to positive by flipping them between numerator and denominator.
• Combine like terms by adding exponents when they have the same base.

In detail, you get:- Move \(x^{-3}\) from the numerator to the denominator and \(y^{-5}\) from the denominator to the numerator.- Combine \(x^{-3}\) and \(x^2\) as \(x^{-3+2} = x^{-1}\).Resulting in a cleaner, simplified fraction ready for further operations.

Positive Exponents

Positive exponents represent how many times to multiply a base by itself. They're easy to handle because they follow straightforward multiplication rules. This exercise requires converting all answers into positive exponents. When dealing with positive and negative exponent combinations, remember:

• Multiply normally if the exponent is positive.
• Converts negative exponents, as they represent reciprocal or inverse values.

For instance, when you reach an expression like \(3x^{-1/3}y^2\), make all exponents positive:- Converting \(x^{-1/3}\) means moving it to the denominator, yielding \(\frac{1}{x^{1/3}}\), making the result easier to interpret in its simplified form.

Rules of Exponents

The rules of exponents are vital when working with expressions that contain variables raised to powers. They provide a consistent method for simplifying and manipulating these expressions. Some key rules include:

• Product Rule: \(a^m \cdot a^n = a^{(m+n)}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{(m-n)}\)
• Power Rule: \((a^m)^n = a^{(mn)}\)
• Zero Exponent Rule: \(a^0 = 1\)
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)

Understanding these rules assists in simplifying complex expressions efficiently. Each offers clarity on how terms interact, providing pathways to simplify even challenging fractions with multiple variables and operations. The step-by-step simplification in this exercise leaned heavily on these rules to transition the expression into its simplest positive exponent form.