Problem 16 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \frac{27}{8} \)

Step 1: Understand Negative Exponent

A negative exponent means that the base is on the wrong side of a fraction line. You need to take the reciprocal of the base to make the exponent positive. Thus, \( \left( \frac{2}{3} \right)^{-3} \) becomes \( \left( \frac{3}{2} \right)^{3} \).

Step 2: Raise the Reciprocal to the Positive Exponent

Once you have the reciprocal \( \left( \frac{3}{2} \right) \), raise it to the positive exponent, which is 3 in this case: \( \left( \frac{3}{2} \right)^3 \).

Step 3: Calculate the Power for Numerator and Denominator

Compute the power for both the numerator and the denominator. For the numerator (3): \( 3^3 = 27 \). For the denominator (2): \( 2^3 = 8 \).

Step 4: Write the Final Result

Combine the results from the numerator and denominator: \( \left( \frac{3}{2} \right)^3 = \frac{27}{8} \). This is the final result of the expression.

Key Concepts

Reciprocal of Fractions

The concept of finding the reciprocal of a fraction is fundamental when dealing with negative exponents. A reciprocal of a fraction is simply a transformation where the numerator and denominator swap places. For any fraction \( \frac{a}{b} \), its reciprocal will be \( \frac{b}{a} \). This step is crucial when converting negative exponents to positive ones because a negative exponent indicates that the base is on the wrong side of a fraction line.
For example, with the expression \( \left( \frac{2}{3} \right)^{-3} \), the negative exponent tells us to take the reciprocal of the fraction \( \frac{2}{3} \), changing it to \( \frac{3}{2} \). Once the reciprocal is determined, the exponent's sign changes from negative to positive. This is because negative exponents cause the base to "flip" into its reciprocal form, essentially moving across the fraction line, thereby reversing its role.

Raising Fractions to a Power

Once the reciprocal has been determined, you will then raise it to the prescribed power. This involves taking the new fraction and multiplying it by itself as many times as indicated by the exponent. For instance, if our fraction is \( \frac{3}{2} \) and the exponent is 3, the process involves multiplying \( \frac{3}{2} \) by itself three times.

• So, \( \left( \frac{3}{2} \right)^3 = \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \).

This results in applying the power to both the numerator and denominator separately. A systematic multiplication confirms the effect of the exponent and helps solidify understanding of this process.
When raising fractions to a power, it's important to carefully carry out the multiplication to avoid mistakes. Each multiplication is vital to reaching the correct result and ensures the full application of the exponent to the fraction.

Power of a Numerator and Denominator

When raising a fraction like \( \frac{a}{b} \) to a power, it's essential to apply the exponent individually to both the numerator and the denominator. This ensures that each part of the fraction is correctly evaluated.
For the specific exercise \( \left( \frac{3}{2} \right)^3 \), we apply the exponent of 3 separately:

• The numerator \(3\): \(3^3 = 3 \times 3 \times 3 = 27\).
• The denominator \(2\): \(2^3 = 2 \times 2 \times 2 = 8\).

After raising both parts to the power, combine the results into a new fraction: \( \frac{27}{8} \).
This process reiterates that each component of the fraction must be addressed, ensuring accurate calculations. Such meticulous attention to detail guarantees that fractions raised to powers are both valid and comprehensible.