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Chapter 1: Problem 7

Evaluate \((0.5)^{2}\) two different ways.

Short Answer

Expert verified
Using two different methods, we found that \((0.5)^{2}\) equals \(0.25\). The first method involves multiplying 0.5 by itself, and the second method requires converting the decimal to a fraction, squaring the fraction, and then converting back to a decimal. Both approaches yield the same result: \(0.25\).

Step by step solution

01

Method 1: Decimal Multiplication

Step 1: Write the exponentiation expression Given the expression, \((0.5)^{2}\), we need to multiply 0.5 by itself. Step 2: Multiply decimals Multiply 0.5 by 0.5: \(0.5 \times 0.5 = 0.25\) Step 3: Conclude Hence, \((0.5)^{2} = 0.25\).
02

Method 2: Conversion to Fraction and Squaring

Step 1: Convert the decimal to a fraction Convert the decimal 0.5 to a fraction: \[ 0.5=\frac{1}{2} \] Step 2: Write the exponentiation expression in terms of the fraction Replace the decimal with the fraction in the expression: \[ \left(\frac{1}{2}\right)^{2} \] Step 3: Square the fraction Square the fraction by squaring the numerator and the denominator: \[ \left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2} = \frac{1}{4} \] Step 4: Convert the fraction to a decimal Convert the fraction back to a decimal: \[ \frac{1}{4}=0.25 \] Step 5: Conclude Hence, \((0.5)^{2} = 0.25\). Both methods result in the same value of 0.25, confirming that \((0.5)^{2} = 0.25\).

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