Verified Answer :

  • Rewrite \(0.09\) as the fraction \(\frac{9}{100}\).
  • Apply the exponentiation rule \(\left(\frac{a}{b}\right)^3 = \frac{a^3}{b^3}\) to get \(\frac{9^3}{100^3}\).
  • Calculate \(9^3 = 729\) and \(100^3 = 1,000,000\).
  • Divide to arrive at the decimal result \(\boxed{0.000729}\).

Explanation

  1. Analyzing the Problem

We need to evaluate the expression \((0.09)^3\). This involves calculating the cube of the decimal \(0.09\). A simple way to perform this calculation is to express the decimal as a fraction first. We know that \(0.09 = \frac{9}{100}\). Thus, the expression becomes \(\left(\frac{9}{100}\right)^3\).

  1. Calculating Numerator and Denominator

Using the laws of exponents, we know that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). Applying this to our expression, we get:
\[\left(\frac{9}{100}\right)^3 = \frac{9^3}{100^3}\]
Now we calculate the numerator and the denominator separately. The numerator is \(9^3 = 9 \times 9 \times 9\). Since \(9 \times 9 = 81\), we have \(81 \times 9 = 729\). The denominator is \(100^3 = 100 \times 100 \times 100 = 1,000,000\).

  1. Final Evaluation

Now we divide the numerator by the denominator:
\[\frac{729}{1,000,000} = 0.000729\]
Therefore, the result of \((0.09)^3\) is \(\boxed{0.000729}\).

Examples

Imagine you are calculating the volume of a very small cube in a scientific experiment. If each side length of the cube measures \(0.09\) meters, you would need to calculate its volume using the formula \(V = s^3\). Substituting the side length, you get \(V = (0.09 \text{ m})^3 = 0.000729 \text{ m}^3\). This conversion of decimals to fractions often helps in keeping calculations accurate when dealing with small physical quantities.

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