Problem 51 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest hundredth. $$\sqrt{\frac{1}{3}}$$

Step-by-Step Solution

Answer

0.58

Step 1: Identify the Expression

The expression given is \( \sqrt{\frac{1}{3}} \). We need to find the square root of the fraction \( \frac{1}{3} \).

Step 2: Calculate the Numeric Value

Use a calculator to find the square root of \( \frac{1}{3} \). First, calculate the decimal equivalent of \( \frac{1}{3} \), which is approximately \( 0.333333 \).

Step 3: Find the Square Root

Now, find the square root of \( 0.333333 \) using the calculator. This gives approximately \( 0.57735 \).

Step 4: Round to the Nearest Hundredth

Round the calculated square root \( 0.57735 \) to the nearest hundredth. Thus, \( 0.57735 \) becomes \( 0.58 \).

Key Concepts

Understanding Approximating Square Roots

Approximating square roots is a key skill in prealgebra, especially when dealing with non-perfect squares. Typically, when you have a square root of a number that is not a perfect square, its result will be an irrational number, meaning it can go on forever without repeating. In our exercise, we're looking at the expression \(\sqrt{\frac{1}{3}}\).
To approximate the square root of a number, you can use a calculator. Begin by converting any fraction, like \(\frac{1}{3}\), into its decimal approximation. Once you have a number to work with, you can use the square root function on the calculator to find the approximate value. This function will help you determine the decimal value closest to the true square root. The key here is to understand that because the square root is irrational, your calculator will give you a rounded version. Therefore, approximating square roots using a calculator is essential for quick, reliable results.

Fraction Conversion to Decimal

Fractions can and often need to be converted into decimals for ease of computation, especially when using a calculator. In the given exercise, we started with the fraction \(\frac{1}{3}\). To convert any fraction into a decimal, you simply divide the numerator by the denominator.
- For \(\frac{1}{3}\), 1 divided by 3 equals approximately 0.333333, repeating indefinitely.

• Begin with simple fractions: For example, \(\frac{1}{2}\) becomes 0.5, \(\frac{3}{4}\) becomes 0.75.


• Use these conversions to prepare numbers for further calculations, like taking the square root.

Converting fractions to decimals helps in simplifying expressions and in using calculators effectively, as many only accept decimal entries. This process ensures that mathematical operations can be performed more smoothly.

Rounding Decimals to Hundredths

Rounding is a method of estimating numbers to a desired degree of precision, often determined by the context of a problem. In this exercise, after finding the approximate value of the square root, which was 0.57735, we round it to the nearest hundredth.

• To do so, look at the third decimal place (thousandths place) to determine whether the second decimal place (hundredths) should be rounded up or remain the same.


• In our case, since the third decimal place is 7, which is greater than 5, we round the second decimal (7) up to 8, resulting in 0.58.

Rounding to the nearest hundredth is common in many math problems and real-world scenarios to ensure numbers are manageable but still accurate. Understanding how to round correctly allows for precise estimation and better communication of numerical information.