Problem 49 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The approximate value is 0.58.

Step 1: Identify the Expression

The expression we need to approximate is \( \frac{\sqrt{3}}{3} \). We need to find the square root of 3 and then divide it by 3, rounding the result to the nearest hundredth.

Step 2: Calculate the Square Root

Using a calculator, find the square root of 3 which is approximately 1.73205.

Step 3: Perform the Division

Take the result from the previous step and divide it by 3: \( \frac{1.73205}{3} \approx 0.57735 \).

Step 4: Round to the Nearest Hundredth

Round 0.57735 to the nearest hundredth. Look at the thousandth place, 7, which is greater than 5, so we round up. Therefore, the rounded result is 0.58.

Key Concepts

Square Root Approximation

Approximating square roots is a crucial skill, especially in prealgebra. Since square roots often result in irrational numbers, these can't always be written down exactly, but instead are approached with approximations. For example, when you take the square root of a number like 3, you don't get a clean number like 2 or 4. Instead, you get approximately 1.73205. This process is simple but requires precision, especially if you're doing it manually:

• First, identify the number you need the square root of. In our example, it's 3.
• Use a calculator to find the square root; most calculators have this function, often labeled as "√" or "sqrt."
• The calculator will provide an approximate decimal value, which you can then use for further calculations.

Understanding square root approximation helps in tackling more complex math problems and develops a strong foundation for learning more about irrational numbers.

Dividing Decimals

Dividing decimals involves several straightforward steps, but can be confusing without practice. It's a fundamental concept for dealing with any arithmetic involving decimal numbers, like in our example where we divided an approximate square root by 3. Here's how you do it with ease:

• When you divide, align the numbers like a standard division problem. Place the number you're dividing on the inside of the division symbol.
• If there’s a decimal in the divisor (number outside), you usually want to move it out by multiplying both sides by 10 until it’s a whole number. This ensures you are dividing properly.
• Perform the division as usual, using long division techniques or a calculator for accuracy.
• In our exercise, dividing 1.73205 by 3 gives us about 0.57735.

Practice will make this process quick and easy, especially when handling more complex problems involving decimals.

Rounding to the Nearest Hundredth

Rounding decimals to a specific place value, such as the nearest hundredth, is a skill that simplifies numbers to make them easier to work with while giving an approximate value. Here are the key steps to rounding correctly:

• Firstly, identify the digit at the hundredth place. In our example, it was 0.57, focusing on the "7" in 0.57735.
• Next, look at the digit directly after the hundredth place, which is the thousandths place. Here, it is a 7.
• If the thousandth digit is 5 or higher, round the hundredth place up. If it's lower than 5, keep the hundredth place as it is.
• In 0.57735, since the thousandths digit is 7, it rounds up to 0.58.

Rounding helps simplify complex decimals to communicate clearer and faster, especially in practical, real-world applications like money and measurements.