Problem 26 - 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 \(-\frac{27}{64}\).

Step 1: Understand the Expression

The expression is \( \left(-\frac{3}{4}\right)^{3} \). This means we need to multiply \(-\frac{3}{4}\) by itself three times.

Step 2: Calculate the First Multiplication

First, multiply \(-\frac{3}{4}\) by \(-\frac{3}{4}\): \[\left(-\frac{3}{4}\right) \times \left(-\frac{3}{4}\right) = \frac{9}{16}\]The negative signs cancel each other out, resulting in a positive value.

Step 3: Multiply the Result by the Base Again

Now multiply the result \(\frac{9}{16}\) by \(-\frac{3}{4}\) again:\[\frac{9}{16} \times \left(-\frac{3}{4}\right) = -\frac{27}{64}\]Here, the positive sign multiplied by a negative sign yields a negative result.

Step 4: Verify with a Calculator

Using a calculator, input \((-3/4)^3\) and verify that the result is indeed \(-\frac{27}{64}\). This confirms our calculation is correct.

Key Concepts

Negative Fractions

A negative fraction is a fraction where the numerator or the denominator is negative. In our exercise, we have \(-\frac{3}{4}\). This simply means you have -3 divided by 4. Negative fractions follow the same operational rules as whole numbers, except they include the rules of fractions too.

When multiplying fractions, two negatives will cancel out and make a positive. This is why, in Step 2, when \(-\frac{3}{4}\) is multiplied by itself, the result is positive \(\frac{9}{16}\).

In contrast, when a positive number (like \(\frac{9}{16}\)) is multiplied by a negative fraction (like \(-\frac{3}{4}\)), the product is negative. Keep in mind:

• Negative x Negative = Positive
• Positive x Negative = Negative

Multiplication

Multiplication of fractions involves numerators being multiplied with numerators, and denominators with denominators. Let's clarify the steps with the expression \(\left(-\frac{3}{4}\right)^3\):

• First multiplication: \(\left(-\frac{3}{4}\right) \times \left(-\frac{3}{4}\right) = \frac{9}{16}\). The multiplicative process of two negative fractions results in a positive fraction because combining two negative signs makes a positive.


• Second multiplication: The result \(\frac{9}{16}\) is positive, and you multiply it by another negative fraction \(-\frac{3}{4}\), resulting in \(-\frac{27}{64}\).

Here are some tips for multiplying fractions:

• Multiply the numerators: \(3 \times 3 = 9\).
• Multiply the denominators: \(4 \times 4 = 16\).
• Reduce fractions if needed after multiplication.

Calculator Verification

After evaluating expressions by hand, using a calculator for verification ensures the accuracy of your results, especially when dealing with complex operations like negative fractions and powers.

Follow these steps for calculator verification:

• Input the fraction as a decimal, if necessary. Use parentheses to ensure correct order of operations.
• For \(\left(-\frac{3}{4}\right)^3\), you might enter it as \(-0.75^3\) in a calculator.
• Check if the calculator gives you \(-\frac{27}{64}\) or its decimal equivalent \(-0.421875\)

Verification builds confidence in your manual work. If the result matches, you know your understanding of the process is solid.