Question

\(\left(\frac{2}{5}\right)^3\) A. \(\frac{6}{5}\) B. \(\frac{8}{5}\) C. \(\frac{6}{15}\) D. \(\frac{8}{125}\)

Short answer

\(\left(\frac{2}{5}\right)^3 = \frac{8}{125}\). The correct answer is D.

Step-by-step solution

Understanding the Problem

The problem asks to compute \(\left(\frac{2}{5}\right)^3\) and choose the correct option from the list.

Applying the Exponentiation Rule

Raise both the numerator and the denominator to the power of 3: \(\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3}\).

Calculating the Powers

Compute \(2^3 = 8\) and \(5^3 = 125\). So, \(\left(\frac{2}{5}\right)^3 = \frac{8}{125}\).

Choosing the Correct Answer

Compare the computed result with the given options. The correct answer is \(\frac{8}{125}\), which corresponds to option D.

Key concepts

raising fractions to a power

When we are asked to raise a fraction to a power, we are essentially performing exponentiation on both the numerator and the denominator. This means if we have a fraction \(\frac{a}{b}\) and we need to raise it to the power of \(n\), we will raise both \(a\) and \(b\) to the power of \(n\).
For example, with \( \frac{2}{5} \) raised to the power of 3, we calculate \[ \left( \frac{2}{5} \right)^3 = \frac{2^3}{5^3} \].
This rule is important because it tells us exactly what steps to follow. It simplifies our process by breaking it down into manageable pieces.

numerator and denominator manipulation

When manipulating the numerator and denominator separately during fraction exponentiation, we follow specific steps:

• Identify the numerator and denominator of your fraction, say \( \frac{a}{b} \).
• Raise the numerator \( a \) to the given power \( n \), which gives us \( a^n \).
• Raise the denominator \( b \) to the same power \( n \), resulting in \( b^n \).

Following these steps for \(\frac{2}{5}\) raised to the power 3, we have:

• \( 2^3=8 \)
• \( 5^3=125 \)

Thus, \( \frac{2^3}{5^3} = \frac{8}{125} \). This separation ensures clarity and accuracy in our calculations. Remember, the same power is applied to both parts of the fraction!

solving step-by-step math problems

Breaking down math problems into smaller steps can make them easier to solve:

• Step 1: Understand the problem. For instance, compute \( \left( \frac{2}{5} \right)^3 \).
• Step 2: Apply the exponentiation rule. Raise both the numerator and the denominator to the power of 3: \(\frac{2^3}{5^3}\).
• Step 3: Calculate each power separately. Compute \ 2^3 = 8 \ and \ 5^3 = 125 \.
• Step 4: Combine your results. The fraction becomes \(\frac{8}{125}\).
• Step 5: Choose the correct answer comparing it with the given options.

Following these steps, we find that option D, \( \frac{8}{125} \), is the correct answer.
Systematically approaching problems like this can make math feel accessible and manageable.