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

Question

Evaluate the expression. $$ \left(-\frac{3}{4}\right)^{2} $$

Step-by-Step Solution

Answer

The given expression is \(\left(-\frac{3}{4}\right)^2\). Applying the exponent rule for fractions and evaluating, we get \(\frac{(-3)^2}{4^2} = \frac{9}{16}\).

Step 1: Identify the base and exponent

The given expression is \(\left(-\frac{3}{4}\right)^2\). Here, the base is \(-\frac{3}{4}\), and the exponent is \(2\).

Step 2: Apply the exponent rule for fractions

When a fraction is raised to an exponent, we can raise both the numerator and the denominator to that exponent separately. In other words, \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). We will apply this rule to our expression: \(\left(-\frac{3}{4}\right)^2 = \frac{(-3)^2}{4^2}\)

Step 3: Evaluate the expressions in the numerator and denominator

Now it's time to calculate the powers in the numerator and denominator: \(\frac{(-3)^2}{4^2} = \frac{9}{16}\) Our final expression is the evaluated form of the given expression, which is \(\frac{9}{16}\).

Key Concepts

Exponent Rules

Understanding exponent rules is critical in simplifying expressions, especially when working with fractions. Exponents, also known as powers, are a shorthand notation to represent repeated multiplication. When dealing with exponents in expressions, there are several rules that help us simplify calculations:

• Product of Powers Rule: This states that when multiplying powers with the same base, you can add the exponents, such as: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power Rule: When you raise a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• Power of a Product Rule: When you raise a product to a power, you raise each factor to the power separately: \((ab)^n = a^n \cdot b^n\).
• Power of a Quotient Rule: When using a fraction as the base, apply the exponent to both the numerator and the denominator separately: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). This last rule is crucial when dealing with fractions like in our example.

Numerator and Denominator

In fractions, the numerator and denominator play distinct roles. The numerator is the top part of a fraction and represents how many parts we have or are considering. The denominator, found at the bottom, shows how many equal parts the whole is divided into.
Let's consider the expression \(\left(-\frac{3}{4}\right)^2\). Here, the fraction's numerator is \(-3\) and the denominator is \(4\). Understanding these components is essential because when a fraction is raised to an exponent, both the numerator and the denominator are raised to that exponent separately.
When the fraction \(-\frac{3}{4}\) is raised to the power of 2, the rule \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) is applied. This means \((-3)^2\) for the numerator and \(4^2\) for the denominator.

Fraction Evaluation

Fraction evaluation involves performing the necessary arithmetic operations to simplify or calculate the value of an expression. In the case of the exponentiated fraction \(\left(-\frac{3}{4}\right)^2\), we initially separate it per the exponentiation rules.

• Calculate the Numerator: By evaluating \((-3)^2\), we find that multiplying \(-3\) by itself yields a positive result, \(9\), because multiplying two negative numbers results in a positive number.
• Calculate the Denominator: Similarly, \(4^2\) is simply \(4 \times 4 = 16\).

Thus, the evaluated fraction is \(\frac{9}{16}\). This process demonstrates how each component, numerator, and denominator, is independently raised to the power, and the results are combined to yield the final evaluated expression.