Problem 52 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest hundredth. $$\sqrt{\frac{1}{2}}$$

Step-by-Step Solution

Answer

The approximate value of \( \sqrt{\frac{1}{2}} \) is 0.71.

Step 1: Understand the Expression

We need to approximate the expression \( \sqrt{\frac{1}{2}} \) to the nearest hundredth. This involves calculating the square root of \( \frac{1}{2} \).

Step 2: Calculate 1/2

First, understand that \( \frac{1}{2} \) is equal to 0.5 in decimal form. This conversion makes it easier to use a calculator for the next step.

Step 3: Use a Calculator to Find the Square Root

Use a calculator to find \( \sqrt{0.5} \). Enter 0.5 and press the square root function button on the calculator.

Step 4: Write the Answer

The calculator should give you approximately 0.707106781. Round this number to the nearest hundredth, which results in 0.71.

Key Concepts

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, produces the original number. Think of it as the opposite of squaring a number. For instance, the square of 3 is 9, so the square root of 9 is 3. This concept is particularly useful in a variety of mathematical problems, including geometry and algebra. When working with fractional or decimal numbers, like \( \frac{1}{2} \) or 0.5, finding the square root is just as straightforward. Simply treat the decimal as the number to be root squared. Calculators simplify this process, ensuring accuracy. They typically have a square root function that handles the calculation for you, providing precise results. This is especially beneficial when you're tasked with finding approximate values, as you can quickly enter the number and retrieve the square root.

Calculating Decimals

Decimals are just another way of representing fractions. They provide a way to show numbers that are less than complete whole numbers, making them especially useful in calculations requiring precision, such as square roots. For example, \( \frac{1}{2} \) is equal to the decimal 0.5. Converting fractions to decimals can often make mathematical operations more manageable. When calculating, dividing the numerator by the denominator converts the fraction into a decimal. Calculators and other tools often require input in decimal form. By presenting numbers in decimals, you facilitate seamless arithmetic operations like addition, subtraction, and finding square roots. Make sure to handle them accurately for the best results.

The Art of Rounding Numbers

Rounding numbers is a mathematical approach used to simplify figures, making them easier to work with, particularly in approximation and estimation. In our case, after calculating the square root of 0.5, the result was approximately 0.707106781. Rounding to the nearest hundredth means looking at the third decimal place. If it's 5 or above, you'll round up the second decimal place by one. If it's below 5, you leave the second decimal as is. Thus, from 0.707106781, you'd round up to 0.71. Here’s a simple way to think about rounding:

• Locate the decimal place you’re interested in (hundredths, tenths, etc.).
• Check the number right after it.
• If that number is 5 or more, increase the target number by one.
• If it's less than 5, keep the target number the same.

Rounding not only provides more digestible numbers but often helps in obtaining quick approximations, especially in everyday math scenarios.