Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

a. \(\left(2^{-1}\right)^{3} = \dfrac{1}{8}\) b. \(\left(3^{-2}\right)^{3} = \dfrac{1}{729}\)

Step 1: a. Solving the expression (2^{-1})^{3}

Step 1: Apply the power of a power rule We have \(\left(2^{-1}\right)^{3}\). Using the power of a power rule, we can rewrite this expression as \(2^{-3}\). Step 2: Simplify the expression Since \(2^{-3} = \dfrac{1}{2^3}\), let's simplify the denominator: \(\dfrac{1}{2^3} = \dfrac{1}{8}\). The expression \(\left(2^{-1}\right)^{3}\) simplifies to \(\dfrac{1}{8}\).

Step 2: b. Solving the expression (3^{-2})^{3}

Step 1: Apply the power of a power rule We have \(\left(3^{-2}\right)^{3}\). Using the power of a power rule, we can rewrite this expression as \(3^{-6}\). Step 2: Simplify the expression Since \(3^{-6} = \dfrac{1}{3^6}\), let's find the denominator: \(\dfrac{1}{3^6} = \dfrac{1}{729}\). The expression \(\left(3^{-2}\right)^{3}\) simplifies to \(\dfrac{1}{729}\).

Key Concepts

Power of a power rule

Understanding powers can be made simpler with specific rules like the "power of a power rule." This rule tells us that when you raise an exponent to another exponent, you multiply the exponents. For example, if you have an expression \(b^{m}\)\(\)^n\, you can simplify it to \(b^{m \cdot n}\). Let's take a closer look. Given a scenario where you have \(\left(2^{-1}\right)^3\), using the power of a power rule, you multiply the exponents: \(-1 \times 3 = -3\). Thus, the simplified expression becomes \(2^{-3}\). This rule saves time and effort in calculations, helping us solve complex expressions more easily.

Negative exponents

Negative exponents can be a bit intimidating at first, but they follow a simple rule: they indicate a reciprocal. If you have a base with a negative exponent, like \(b^{-n}\), it becomes \(\frac{1}{b^n}\). Consider an example: simplifying \(2^{-3}\). By applying the negative exponent rule, it transforms to \(\frac{1}{2^3}\). This tells you to flip the base to its reciprocal in fraction form. The concept appears widely in mathematics because it helps describe very small numbers efficiently. Becoming comfortable with negative exponents encourages confidence in tackling more advanced math problems.

Simplification of expressions

The purpose of simplifying expressions is to make them easier to work with and understand. When dealing with expressions involving exponents and fractions, simplification transforms them into a more manageable form.For instance, let's simplify \(\frac{1}{2^3}\). Calculate the denominator: \(2^3 = 8\), hence \(\frac{1}{2^3} = \frac{1}{8}\). Similarly, with \(\frac{1}{3^6}\), compute \(3^6 = 729\), simplifying it to \(\frac{1}{729}\). Simplification helps make math operations smoother. It helps in making complex expressions less cumbersome and allows for clearer insight into the problem by reducing it to a more fundamental form.