Problem 22 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ \left(-\frac{2}{3}\right)^{-3} $$

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{27}{8}\).

Step 1: Understand the Negative Exponent

A negative exponent indicates taking the reciprocal of the base with a positive exponent. For example, if you have a number \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\).

Step 2: Apply the Negative Exponent Rule

For the expression \(\left(-\frac{2}{3}\right)^{-3}\), take the reciprocal of \(-\frac{2}{3}\) and change the exponent to positive, giving us \(\left(-\frac{3}{2}\right)^3\).

Step 3: Expand the Positive Exponent

Now calculate \(\left(-\frac{3}{2}\right)^3\), which involves multiplying \(-\frac{3}{2}\) by itself three times: \(-\frac{3}{2} \times -\frac{3}{2} \times -\frac{3}{2}\).

Step 4: Perform the Multiplication

Multiply the numerators: \(-3 \times -3 \times -3 = -27\). Multiply the denominators: \(2 \times 2 \times 2 = 8\). So, \(\left(-\frac{3}{2}\right)^3 = \frac{-27}{8}\).

Step 5: Verify with a Calculator

Use a calculator to compute \(\frac{-27}{8}\) and verify that the result is consistent with our hand-calculated answer. Input "(-2/3)^-3" to ensure it corresponds with \(-3/2\) raised to the third power, which results in \(-27/8\).

Key Concepts

Reciprocal

The concept of a reciprocal is quite interesting and essential when dealing with negative exponents. A reciprocal of a number is simply 1 divided by that number. It essentially "flips" the fraction. For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). Thus, the roles of the numerator and the denominator are swapped.

When you have an expression with a negative exponent, such as \( a^{-n} \), you are directed to find the reciprocal. This is because negative exponents indicate the inversion of the base's powers. So, \( a^{-n} = \frac{1}{a^n} \).

Let’s apply this to \( \left(-\frac{2}{3} \right)^{-3} \). Here, the negative exponent means we take the reciprocal of \( -\frac{2}{3} \) to form \( -\frac{3}{2} \). Then, you raise this reciprocal to a positive power. This step is crucial for transforming negative exponents into a more manageable positive exponent form.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it's really about following a straightforward process. When you multiply fractions, you multiply the numerators together and then the denominators together.

For instance, if multiplying \( \frac{a}{b} \) by \( \frac{c}{d} \), the result would be \( \frac{a \times c}{b \times d} \). This method provides a simple way to handle fractions across various mathematical problems, including calculating powers of fractions.

In the case of \( \left(-\frac{3}{2}\right)^3 \), apply this rule by multiplying \(-\frac{3}{2} \times -\frac{3}{2} \times -\frac{3}{2}\). This means:

• Multiplying the numerators: \(-3 \times -3 \times -3 = -27\)
• Multiplying the denominators: \(2 \times 2 \times 2 = 8\)

Thus, the fraction \( \frac{-27}{8} \) is calculated by multiplying these components. Understanding this will vastly simplify working with fractional exponents and multiplications.

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. The exponent denotes how many times the base is to be multiplied by itself. For instance, \( a^n \) means "multiply \( a \) by itself \( n \) times."

With fractional bases, like \( \left(-\frac{2}{3}\right)^{-3} \), understanding expansion through exponentiation is important. First, handle the negative exponent by finding the reciprocal. Then, apply the positive exponent normally:

• This involves repeating multiplication: \(-\frac{3}{2} \times -\frac{3}{2} \times -\frac{3}{2}\)
• The power of three, in this situation, signifies three successive multiplications of the reciprocal.

Exponentiation showcases the power of repeated multiplication, whether with simple integers or more complex fractions. This helps to streamline calculations in algebra and beyond, making sense of even the most complicated exponential expressions.