Verified Answer :

  • Interpret the exponentiation as the product \(0.5 \times 0.5 \times 0.5\).
  • Convert the decimal to a fraction: \(0.5 = \frac{1}{2}\).
  • Calculate the power of the fraction: \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\).
  • Convert the fraction to the decimal \(\boxed{0.125}\).

Explanation

  1. Understanding the Expression

We are asked to evaluate the expression \((0.5)^3\). The expression involves the number \(0.5\) raised to the third power. Exponentiation means multiplying the base, in this case \(0.5\), by itself the number of times indicated by the exponent, which is \(3\). So, we need to calculate \(0.5 \times 0.5 \times 0.5\).

  1. Converting to Fractions

To make the calculation more intuitive, we can convert the decimal \(0.5\) into its fractional form, which is \(\frac{1}{2}\). The expression then becomes \(\left(\frac{1}{2}\right)^3\). According to the rules of exponents, this is equivalent to \(\frac{1^3}{2^3}\).

  1. Calculating the Resulting Fraction

Now, we calculate the numerator and the denominator separately:

  • The numerator is \(1^3 = 1 \times 1 \times 1 = 1\).
  • The denominator is \(2^3 = 2 \times 2 \times 2 = 8\).

Thus, \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\).

  1. Final Calculation

Finally, we convert the fraction \(\frac{1}{8}\) back to a decimal. By performing the division \(1 \div 8\), we get \(0.125\). Thus, \((0.5)^3 = \boxed{0.125}\).

Examples

Exponentiation is widely used in finance and science. For instance, if you are calculating compound interest over three time periods with a growth rate of 50%, you might represent the final value factor as \((1 + 0.5)^3\). In physics, volume calculations often involve cubing a length, such as finding the volume of a cube with a side length of \(0.5\) meters, which is \(0.5^3 = 0.125\) cubic meters. Understanding how to handle these powers helps in predicting growth and calculating spatial dimensions accurately.

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