Problem 31 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ \left(\frac{3}{4}\right)^{-3} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{64}{27} \).

Step 1: Understand the Negative Exponent

The expression \( \left(\frac{3}{4}\right)^{-3} \) indicates that you need to take the reciprocal of \( \frac{3}{4} \) and then raise it to the power of 3. This is because a negative exponent means to invert the base and change the sign of the exponent to positive.

Step 2: Find the Reciprocal

The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). So, \( \left(\frac{3}{4}\right)^{-3} = \left(\frac{4}{3}\right)^3 \).

Step 3: Raise the Reciprocal to the Power of 3

Now calculate \( \left(\frac{4}{3}\right)^3 \). This involves raising both the numerator and the denominator to the power of 3: \[\left(\frac{4}{3}\right)^3 = \frac{4^3}{3^3} = \frac{64}{27}.\]

Step 4: Finalize the Solution

Hence, \( \left(\frac{3}{4}\right)^{-3} = \frac{64}{27} \). You can verify this by calculating \( \frac{64}{27} \) using a calculator to ensure the accuracy of your manual calculation.

Key Concepts

Reciprocal

A reciprocal is quite simply a fraction flipped upside down. In mathematical terms, the reciprocal of a number is one divided by that number. For fractions, this means swapping the numerator and the denominator.
For example:

• The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
• The reciprocal of \( \frac{7}{2} \) would be \( \frac{2}{7} \).

When dealing with negative exponents, finding the reciprocal is the first step. A negative exponent, like in \( \left(\frac{3}{4}\right)^{-3} \), instructs us to take the reciprocal of the base before raising it to a positive power. This flips \( \frac{3}{4} \) to \( \frac{4}{3} \), preparing it for the next step in the process.

Exponentiation

Exponentiation refers to the operation of raising a number or expression to a power, which means multiplying the number, called the base, by itself as many times as indicated by the exponent. When the base is a fraction, each part of the fraction (numerator and denominator) is raised to the power individually.

For instance, when you have \( \left(\frac{4}{3}\right)^3 \):

• Raise the numerator: \( 4^3 = 4 \times 4 \times 4 = 64 \).
• Raise the denominator: \( 3^3 = 3 \times 3 \times 3 = 27 \).

This calculation results in \( \frac{64}{27} \), representing \( \left(\frac{4}{3}\right)^3 \).

Exponentiation is a key concept in mathematics as it allows us to express repeated multiplication compactly. Understanding how to apply exponents to fractions is crucial for simplifying and solving more complex equations.

Fractions

Fractions are a way of expressing numbers that are not whole. They consist of two parts: a numerator and a denominator. The numerator represents how many parts you have, and the denominator shows the total number of equal parts the whole is divided into.

For example, in the fraction \( \frac{3}{4} \):

• The numerator is 3, meaning you have 3 parts.
• The denominator is 4, meaning the whole is divided into 4 parts.

Fractions can represent quantities less than one or greater than one, depending on whether the numerator is smaller or larger than the denominator.

In operations such as exponentiation, each part of the fraction is dealt with separately. This ensures the accuracy and consistency of results. Grasping the basic principles of working with fractions is essential for understanding more advanced mathematical concepts, such as operations involving negative exponents. By learning how to manipulate fractions, you set a foundation for tackling diverse and interesting mathematical problems.