Evaluate. Write your answer as a decimal or whole number.
$(0.05)^3=$
The problem asks us to evaluate the expression \((0.05)^3\). This involves raising the decimal number \(0.05\) to the power of \(3\), which is equivalent to multiplying the number by itself three times: \(0.05 \times 0.05 \times 0.05\). To make the calculation easier, we can convert the decimal into a fraction. We know that \(0.05 = \frac{5}{100} = \frac{1}{20}\). Therefore, the expression becomes \(\left(\frac{1}{20}\right)^3\).
Applying the exponent to the fraction \(\left(\frac{1}{20}\right)^3\) gives \(\frac{1^3}{20^3}\). Calculating the numerator, we have \(1 \times 1 \times 1 = 1\). For the denominator, we calculate \(20^3 = 20 \times 20 \times 20 = 400 \times 20 = 8000\). So, the expression in fractional form is \(\frac{1}{8000}\).
To express the result as a decimal, we divide the numerator by the denominator: \(\frac{1}{8000}\). We can rewrite this as \(\frac{1}{8} \times \frac{1}{1000}\). We know that \(\frac{1}{8} = 0.125\). Thus, \(\frac{0.125}{1000} = 0.000125\). Therefore, \((0.05)^3 = 0.000125\).
Understanding how to handle powers of decimals is useful in fields like finance and science, for instance, when calculating compound interest over very small time increments or determining the probability of a sequence of rare independent events. If an event has a probability of 0.05 (or 5%), the probability of it occurring three times in a row is \((0.05)^3 = 0.000125\), helping you assess the likelihood of unlikely occurrences in real-world scenarios.