Problem 48 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest hundredth. $$5 \sqrt{3}$$

Step-by-Step Solution

Answer

The approximate value of \(5 \sqrt{3}\) is 8.66.

Step 1: Understand the Operation

The problem requires us to approximate the expression \(5 \sqrt{3}\) to the nearest hundredth. We need to multiply 5 by the square root of 3.

Step 2: Calculate the Square Root

First, find the square root of 3. Using a calculator, compute \( \sqrt{3} \approx 1.73205 \).

Step 3: Perform the Multiplication

Now, multiply 5 by the approximate value of \( \sqrt{3} \): \[5 \times 1.73205 = 8.66025\]

Step 4: Round to the Nearest Hundredth

Round the result from the multiplication, 8.66025, to the nearest hundredth. The hundredth digit is 6, and since the thousandth digit is 0, the hundredth stays the same: \[8.66\]

Key Concepts

Square Roots

The concept of a square root involves finding a number that, when multiplied by itself, will give you the original number under the square root sign. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9.
In mathematical notation, this is expressed as \( \sqrt{9} = 3 \). Square roots are common in calculations, especially when you need precise measurements or calculations. Breaking down square roots allows you to understand numbers in their simplest forms.
When dealing with numbers that aren't perfect squares (like 3), we get decimal expressions by using calculators. For instance, \( \sqrt{3} \approx 1.73205 \). You seldom need all these digits, and often a simplified form suffices.

Approximation

Approximation is about finding a value that is close enough to the right answer. Often, it's unnecessary to keep lengthy decimal numbers in everyday calculations. For numbers like \( \sqrt{3} \), we use approximation to express its value in a more digestible form. This is useful in both simple and complex calculations where exact precision isn't required.
For instance, when we calculated \( 5 \times \sqrt{3} \) to be approximately 8.66025, only using the number to a suitable degree of precision allows you to manage calculations easily without losing significant accuracy. Using shorter numbers helps avoid errors that longer sequences of digits might otherwise introduce.

Rounding Numbers

Rounding numbers simplifies them to make calculations easier and more manageable. When rounding numbers to the nearest hundredth, you look at the number in the thousandth place to decide if you'll round the hundredth place up or keep it as is.
For example, if you are rounding the number 8.66025 to the nearest hundredth, you check the digit in the thousandth place (which is 0 in this case). Since it's less than 5, you keep the hundredth digit as is, resulting in the rounded number of \( 8.66 \).

• If the digit is 5 or more, you increase the hundredth place by one.
• If the digit is less than 5, you don't change the hundredth place.

Such rounding off is effective in maintaining a balance between accuracy and simplicity, ensuring your figures are easy to work with while still accurate enough for most purposes.