Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. a. \(\left[\left(-\frac{1}{2}\right)^{3}\right]^{-2}\) b. \(\left[\left(-\frac{1}{3}\right)^{2}\right]^{-3}\)

Step-by-Step Solution

Answer

a. \(\left[\left(-\frac{1}{2}\right)^{3}\right]^{-2} = 64\) b. \(\left[\left(-\frac{1}{3}\right)^{2}\right]^{-3} = 729\)

Step 1: Solve the expression inside the brackets

Raise \(-\frac{1}{2}\) to the power of \(3\): \[\left(-\frac{1}{2}\right)^3 = -\frac{1^3}{2^3} = -\frac{1}{8}\]

Step 2: Evaluate the negative exponent

Apply the negative exponent rule for \(\left[-\frac{1}{8}\right]^{-2}\): \[\left(-\frac{1}{8}\right)^{-2} = \left(\frac{1}{-\frac{1}{8}}\right)^2 = \left(\frac{1}{-1/8}\right)^2 = \left(-8\right)^2\]

Step 3: Calculate the final result

Square the number inside the brackets: \[\left(-8\right)^2 = (-8)(-8) = 64\] The expression equals \(64\). b. Evaluate the expression: \(\left[\left(-\frac{1}{3}\right)^{2}\right]^{-3}\)

Step 1: Solve the expression inside the brackets

Raise \(-\frac{1}{3}\) to the power of \(2\): \[\left(-\frac{1}{3}\right)^2 = \left(-\frac{1}{3}\right) \times \left(-\frac{1}{3}\right) = \frac{1}{9}\]

Step 2: Evaluate the negative exponent

Apply the negative exponent rule for \(\left[\frac{1}{9}\right]^{-3}\): \[\left(\frac{1}{9}\right)^{-3} = \left(\frac{1}{\frac{1}{9}}\right)^3 = \left(9\right)^3\]

Step 3: Calculate the final result

Cube the number inside the brackets: \[9^3 = 9 \times 9 \times 9 = 729\] The expression equals \(729\).

Key Concepts

Negative Exponents

In mathematics, negative exponents can initially seem confusing, but once you break them down, they become quite easy to understand. A negative exponent indicates that instead of multiplying, you are actually dividing, or finding the reciprocal of the base raised to the opposite positive exponent.

For instance, if you have an expression like \( x^{-n} \), this is equivalent to \( \frac{1}{x^n} \). Rather than powering up, you power down by taking the reciprocal.

Consider the step where we dealt with \( \left(-\frac{1}{8}\right)^{-2} \). Here, \( -\frac{1}{8} \) has a negative exponent. Converting to positive, we take the reciprocal: \( \left( \frac{1}{-\frac{1}{8}} \right)^2 \). The expression inside is now \(-8\), making handling of exponents straightforward.

Fractional Exponents

Fractional exponents represent both roots and powers, acting as a bridge between multiplication and division of powers. When you encounter a fractional exponent, it might look like \( a^{m/n} \). Here, the numerator \( m \) denotes the power, while the denominator \( n \) indicates the root.

Think of \( a^{1/n} \) as the \( n^{th} \) root of \( a \). If you have \( a^{3/2} \), for example:

• Find the square root of \( a \) (because \( 2 \) is the root)
• Then raise it to the power of \( 3 \) (since \( 3 \) is the power)

Fractional exponents are a handy way of expressing problems that involve roots and powers neatly in one term.

Mathematical Operations

Mathematical operations with exponents follow certain rules that make manipulation systematic and manageable:

Base Rules:
A consistent base in the expression allows simplification using exponent laws. For example, with a base of \(-\frac{1}{2}\) raised to various powers, handle each operation step by step.

Order of Operations:
Recall PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) when evaluating expressions. Always start with operations inside parentheses or brackets, then deal with exponents.

In the examples from the problem, each step involved:

• Calculating the power inside the brackets first
• Applying the negative exponent to inverse or take reciprocal of the base
• Simplifying the result, often by further powers

Practicing these operations builds a solid foundation for tackling more complex algebra and calculus problems.