Verified Answer :

  • Express the decimal \(1.7\) as the fraction \(\frac{17}{10}\).
  • Apply the exponentiation rule \(\left(\frac{17}{10}\right)^2 = \frac{17^2}{10^2}\).
  • Calculate the values of the numerator \(17^2 = 289\) and the denominator \(10^2 = 100\).
  • Divide the result to get the final decimal: \(\frac{289}{100} = \boxed{2.89}\).

Explanation

  1. Representing the Number as a Fraction

To evaluate the expression \((1.7)^2\), we first represent the decimal number \(1.7\) as a fraction. A decimal number with one decimal place can be written as a fraction with a denominator of 10. Thus, \(1.7 = \frac{17}{10}\). The expression we want to evaluate is therefore \(\left(\frac{17}{10}\right)^2\).

  1. Squaring the Numerator and Denominator

Using the rule for squaring a fraction, which states that \(\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}\), we can rewrite our expression as:
\[\left(\frac{17}{10}\right)^2 = \frac{17^2}{10^2}\]
Next, we calculate the square of the numerator and the denominator separately. The numerator is \(17^2 = 17 \times 17 = 289\), and the denominator is \(10^2 = 10 \times 10 = 100\).

  1. Final Calculation and Conversion to Decimal

Now we combine the results from the previous step to get the fraction \(\frac{289}{100}\). To convert this fraction into a decimal, we perform the division by 100, which is equivalent to moving the decimal point two places to the left. Thus, \(\frac{289}{100} = 2.89\). Therefore, \((1.7)^2 = 2.89\).

Examples

Squaring numbers is a fundamental operation used in many real-life applications. For example, if you are measuring the area of a square garden where each side measures 1.7 meters, the total area of the garden is \(1.7 \times 1.7 = 2.89\) square meters. Understanding this simple operation helps in calculating areas, determining growth rates in finance, or solving various physics problems where power or square laws are involved.

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