Problem 47 - Prealgebra

Question

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

Step-by-Step Solution

Answer

11.18

Step 1: Understand the expression

The expression is given as \(5 \sqrt{5}\). This means you need to find the square root of 5 first and then multiply the result by 5.

Step 2: Calculate the square root

Use a calculator to find the square root of 5. The square root of 5 (\(\sqrt{5}\)) is approximately 2.236.

Step 3: Multiply by 5

Now, multiply the result from the previous step by 5. So, \(5 \times 2.236 = 11.18\).

Step 4: Round to the nearest hundredth

After performing the multiplication, the result is 11.18. Since it is already rounded to two decimal places, this is the final answer.

Key Concepts

Approximation

When dealing with irrational numbers, like the square root of 5 (\( \sqrt{5} \)), precise calculations are nearly impossible without advanced computing tools. To simplify, we approximate. Approximating means finding a value close enough to the real answer that is easier to work with.
Numerical approximation simplifies complex calculations by rounding numbers to a specific number of decimal places. For example, \( \sqrt{5} \) is about 2.236067977, an unwieldy number in regular calculations. Instead, we can round this to a more manageable 2.236, or even just 2.24 when the problem asks for rounding to the nearest hundredth.
This makes the number easier to multiply or add in further steps. Remember:

• Look at the third decimal to decide if the second decimal should round up or not.
• If the third digit is 5 or more, round up.
• If it's 4 or less, keep the second decimal as is.

This simple rounding method keeps calculations straightforward while maintaining acceptable accuracy.

Using Calculators

Calculators are powerful tools designed to handle both simple and complicated mathematical calculations. In this exercise, a calculator assists in finding \( \sqrt{5} \), a difficult number to calculate by hand.
Most calculators have a square root function, represented as a \(\sqrt{}\) button. Here’s a simplified approach to using a calculator for this kind of operation:

• Turn on your calculator, and find the \(\sqrt{}\) button.
• Input the number 5, then press the \(\sqrt{}\) function key.
• The calculator will display an approximation of \( \sqrt{5} \).

This process breaks down complex operations into more straightforward steps, ultimately making calculations faster and more accurate. With an accurate approximation of \( \sqrt{5} \) displayed, multiplying it by another number becomes much easier. By mastering calculator functions, students tackle complex problems more efficiently.

Multiplying Decimals

Once you have approximated \( \sqrt{5} \), multiplying by 5 involves decimals. Decimal multiplication requires practice and understanding to ensure precise results.
Here's a simpler way to think about multiplying the rounded number, in this case, 2.236 by a whole number like 5.

• Line up numbers as you usually would in whole number multiplication.
• Ignore the decimal points initially and multiply 2236 by 5.
• Here's the practical part: Count how many total decimal places are present in the numbers you're multiplying. Only the number 2236 has three decimal places.
• Your product, obtained from multiplying 2236 by 5, is 11180.
• Now, re-insert the decimal, moving three places from the right, giving you 11.180.
• Finally, round to the nearest hundredth: check the third decimal place and adjust accordingly, resulting in 11.18.

Knowing these steps ensures you effortlessly multiply decimals, arriving at a precise result nicely rounded for ease of communication. Multiplying decimals becomes manageable with practice, and these steps guide students through any complexity.