Problem 18 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. Write fractional answers in simplest form.\(\left(\frac{-5}{4}\right)^{3}\left(\frac{4}{5}\right)^{2}\)

Step-by-Step Solution

Answer

\(-\frac{5}{4}\)

Step 1: Evaluate the First Expression

Solve the expression in the parentheses first \(\left(\frac{-5}{4}\right)^{3}\). This means, \(-5/4\) is multiplied by itself 3 times. This results in -1.953125 or \(-\frac{125}{64}\).

Step 2: Evaluate the Second Expression

Now solve the second expression, \(\left(\frac{4}{5}\right)^{2}\). This means, \(4/5\) is multiplied by itself 2 times. This equals to 0.64 or \(\frac{16}{25}\).

Step 3: Multiply the Evaluations and Simplify

Multiply the two evaluated expressions \(\left(-\frac{125}{64}\right)\) and \(\left(\frac{16}{25}\right)\). This simplifies to \(-\frac{5}{4}\). So, the final answer in simplest form is \(-\frac{5}{4}\).

Key Concepts

Understanding Fractions

Fractions represent a part of a whole. They consist of a numerator, which is the top number, and a denominator, which is the bottom number. In a fraction, the numerator indicates how many parts you have, while the denominator tells you how many parts make up a whole.
For example, in the fraction \( \frac{3}{4} \), 3 is the numerator, and 4 is the denominator. This fraction represents 3 out of 4 equal parts.
Working with fractions involves multiplying, dividing, adding, or subtracting these numbers. Understanding how to work with fractions is essential in simplifying expressions and solving equations.

• To multiply fractions, multiply the numerators together and the denominators together.
• To divide fractions, multiply by the reciprocal of the divisor.

These fundamental operations will help you manage fractions in exponentiation and simplification.

Demystifying Negative Exponents

Negative exponents might seem confusing at first, but they just represent the process of taking the reciprocal of a number.
When you see a negative exponent, such as \(a^{-n}\), it means you need to take the reciprocal of the base, \(a\), and raise it to the positive exponent, \(n\). Thus, \(a^{-n} = \frac{1}{a^n}\).
This concept is useful when simplifying expressions, especially those involving fractions and exponents.
For example, \(\left(\frac{3}{4}\right)^{-2}\) is equivalent to \(\left(\frac{4}{3}\right)^{2}\), which results in \(\frac{16}{9}\).

• Negative exponents shift the power of a value to its reciprocal.
• This inversion makes performing operations more manageable, as it often helps in repositioning numbers in an equation or expression when simplifying.

Mastering Simplifying Fractions

Simplifying fractions is about expressing a fraction in its simplest form, where the numerator and denominator share no common factor other than 1. This process makes calculations easier and results clearer.
To simplify a fraction, find the greatest common divisor (GCD) of both the numerator and the denominator, and divide both by this number.
For instance, simplifying \(\frac{20}{60}\) involves finding the GCD, which is 20. Dividing both numbers gives \(\frac{1}{3}\).

• Always look for common factors to reduce the fraction to its simplest form.
• This step is crucial after performing operations like addition, subtraction, multiplication, or division to ensure the result is as straightforward as possible.

In our exercise, after performing multiplication, we achieved the simplified form of \(-\frac{5}{4}\) by canceling out common factors where possible, ensuring clarity and precision in the final result.