Problem 22 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ \left(\frac{2}{3}\right)^{-3} $$

Step-by-Step Solution

Answer

\( \left( \frac{2}{3} \right)^{-3} = \frac{27}{8} \).

Step 1: Understand the Negative Exponent

The expression \( \left( \frac{2}{3} \right)^{-3} \) involves a negative exponent, which indicates taking the reciprocal of the base and making the exponent positive. In general, \( a^{-b} = \frac{1}{a^{b}} \). Therefore, rewrite the expression as \( \frac{1}{\left( \frac{2}{3} \right)^{3}} \).

Step 2: Raise the Reciprocal to the Positive Exponent

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

Step 3: Multiply Fractions

To multiply fractions, multiply the numerators together and the denominators together. So, for \( \left( \frac{3}{2} \right)^{3} \): \[ \frac{3 \times 3 \times 3}{2 \times 2 \times 2} = \frac{27}{8} \].

Step 4: Final Result

The expression \( \left( \frac{2}{3} \right)^{-3} \) simplifies to \( \frac{27}{8} \).

Key Concepts

Reciprocal

When you encounter a negative exponent, such as in the expression \( \left( \frac{2}{3} \right)^{-3} \), it signals that you need to take the reciprocal of the base and switch the negative exponent to a positive one. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). The concept of a reciprocal helps us transform negative exponents into a more manageable form.

• Begin by flipping the fraction to find its reciprocal.
• Convert the negative exponent to a positive exponent.

Applying these steps to \( \left( \frac{2}{3} \right)^{-3} \), we first take the reciprocal, getting \( \frac{3}{2} \), and then raise it to the positive power of 3. This process changes our original problem into \( \left( \frac{3}{2} \right)^{3} \), simplifying it into a much more straightforward expression.

Multiplying Fractions

Multiplying fractions is a key skill in mathematics, particularly when dealing with expressions involving exponents. If we return to our transformed expression \( \left( \frac{3}{2} \right)^{3} \), it requires multiplying the fraction \( \frac{3}{2} \) by itself three times.

• Multiply the numerators: \( 3 \times 3 \times 3 = 27 \).
• Multiply the denominators: \( 2 \times 2 \times 2 = 8 \).

This results in the fraction \( \frac{27}{8} \). Remember:

• Keep your fractions simplified, if possible.
• Check for common factors before multiplying, which can simplify your calculations.

This skill is useful not only in this exercise but also in broader algebraic manipulations involving fractions and exponents.

Raising Fractions to a Power

Raising fractions to a power involves multiplying the fraction by itself as many times as the exponent indicates. In the expression \( \left( \frac{3}{2} \right)^{3} \), we are raising the fraction \( \frac{3}{2} \) to the third power. Follow these steps:

• Identify the fraction to be raised to the power.
• Repeat the multiplication of this fraction by itself the number of times indicated by the exponent.

For the fraction \( \frac{3}{2} \), raising it to the power of 3 means multiplying it three times:\[ \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \]This operation gives us \( \frac{27}{8} \). Each multiplication consolidates the fraction's numerators and denominators, respectively. This skill is essential for handling more complex compound fractional expressions cleanly and correctly.