Evaluate. Write your answer as a decimal or whole number.
$(0.1)^2=$
The expression \((0.1)^2\) asks us to square the decimal 0.1. Squaring a number means multiplying the number by itself. Therefore, \((0.1)^2\) is equivalent to \(0.1 \times 0.1\). We can approach this by converting the decimal to a fraction to make the multiplication easier to visualize.
The decimal 0.1 can be written as the fraction \(\frac{1}{10}\). When we raise a fraction to a power, we raise both the numerator and the denominator to that power:
\[\left(\frac{1}{10}\right)^2 = \frac{1^2}{10^2} = \frac{1}{100}\]
This calculation shows that squaring 0.1 is the same as finding one hundredth.
Finally, we convert the fraction \(\frac{1}{100}\) back into decimal notation. Since the denominator is 100, the decimal representation will have two decimal places. Thus, \(\frac{1}{100} = 0.01\). This confirms that \((0.1)^2 = 0.01\).
Understanding how to square small decimals is helpful in geometry and finance. For instance, if you are calculating the area of a small square garden plot that is 0.1 meters by 0.1 meters, the area would be \(0.1 \times 0.1 = 0.01 \text{ m}^2\). Similarly, when calculating interest or growth rates involving small decimal percentages, knowing how to handle decimal multiplication ensures accurate results.