Question

Raise to the power indicated and remove parentheses. $$\left(\frac{2}{3}\right)^{3}$$

Short answer

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

Step-by-step solution

Understanding the Exponent Rule

When raising a fraction to an exponent, the rule states that you must apply the exponent to both the numerator and the denominator separately.

Applying the Exponent to the Numerator

Raise the numerator of the fraction to the power of 3: \(2^3 = 2 \times 2 \times 2 = 8\).

Applying the Exponent to the Denominator

Raise the denominator of the fraction to the power of 3: \(3^3 = 3 \times 3 \times 3 = 27\).

Combining the Results

Combine the results of the numerator and the denominator to complete the exponentiation of the fraction: \(\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}\).

Key concepts

Exponent Rules

Understanding exponent rules is key to dealing with powers in mathematical expressions. Essentially, when you have a number to the power of an exponent, it means you multiply the base (the number being raised to a power) by itself a certain number of times defined by the exponent. For example, when we have expressions like \( 5^3 \), it essentially represents \( 5 \times 5 \times 5 \).

Specifically for fractions, each component of the fraction, the numerator, and the denominator, is raised to the power of the exponent separately. This is what we are doing when we perform the operation \( \left(\frac{2}{3}\right)^3 \). It may help to recall that a fraction is really a form of division, and exponent rules apply just the same to both parts of the division.

Numerical Fractions

Numerical fractions represent parts of a whole. In a fraction like \( \frac{2}{3} \), the upper part—2—is the numerator, indicating how many parts we have. The bottom part—3—is the denominator, revealing into how many parts the whole is divided. When you're dealing with exponentiation of numerical fractions, each part of the fraction is influenced by the exponent.

This process transforms both the numerator and the denominator, modifying the fraction's value while maintaining its nature as a ratio. It's very intuitive when you break it down: you're essentially applying the exponent to two separate whole numbers before recombining them into a new fraction.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form while keeping their values unchanged. With numerical fractions, this often means finding the greatest common factor of the numerator and the denominator and dividing both by that number to simplify. In the context of exponentiation, after the powers have been applied to both parts of the fraction, no further simplification may be possible. This is the case with our example \( \left(\frac{2}{3}\right)^3 = \frac{8}{27} \).

This fraction is already in its simplest form because 8 and 27 have no common factors other than 1. In other situations, you might find that after raising a fraction to a power, you can reduce the resultant fraction to make it simpler. Simplifying expressions makes them easier to work with and can offer a clearer picture of what they represent in the context of the problem.