Problem 81 - Prealgebra

Question

Use a calculator to approximate the square root to the nearest tenth. \(\sqrt{444}\)

Step-by-Step Solution

Answer

The approximate value of \(\sqrt{444}\) to the nearest tenth is 21.1.

Step 1: Understanding the Problem

The problem is asking for an approximation of the square root of 444 to the nearest tenth. This means we need to find a decimal number that is close to the square root and round it to one decimal place.

Step 2: Using a Calculator

Use a calculator to find the square root of 444. Inputting 444 and pressing the square root function should give the result. For this exercise, let's assume the calculator gives us approximately 21.0713075.

Step 3: Rounding to the Nearest Tenth

The number we obtained from the calculator is 21.0713075. To round this to the nearest tenth, look at the first two digits after the decimal point (07). Since the digit in the hundredths place is 7, we round up the tenths place from 0 to 1.

Step 4: Final Approximation

After rounding, the approximate value of the square root of 444 to the nearest tenth is 21.1.

Key Concepts

Rounding Decimals

Rounding decimals helps to simplify numbers. This makes them easier to work with, especially when precision isn't critical. When you round to the nearest tenth, you focus on the first digit after the decimal point, which is known as the tenths place.
To round a decimal to the nearest tenth, follow these easy steps:

• Identify the tenths and hundredths places in the number. For example, in 21.0713075, 0 is in the tenths and 7 is in the hundredths place.
• Look at the digit in the hundredths place. If it is 5 or greater, you round up the tenths place by one. If it is less than 5, keep the tenths place the same.
• Apply this rule to the number. Since 7 is more than 5, 21.0713075 becomes 21.1 when rounded to the nearest tenth.

Understanding how to round properly is a vital skill, particularly in quickly estimating values in day-to-day math or scientific calculations.

Using a Calculator

Using a calculator simplifies complex calculations to save time and reduce errors. Most modern calculators have a function for finding square roots. This is helpful when dealing with numbers that are not perfect squares.
To find the square root of a number using a calculator, simply:

• Enter the number, in this case, 444.
• Press the square root button (usually denoted as \(\sqrt{}\) or something similar).
• Read the result. The calculator might show a long decimal, such as 21.0713075.

This function allows you to find square roots efficiently. Keep in mind that the given result is an approximation unless the number is a perfect square, such as 9 or 16.

Approximating Irrational Numbers

Irrational numbers cannot be expressed as exact fractions or decimals that terminate or repeat. They often appear as square roots. Approximating these numbers means finding a nearby rational number.
Here's how you approach this task:

• The square root of 444 is an example. It doesn't result in a neat, simple number.
• Instead, it gives a long decimal tail (21.0713075) since 444 is not a perfect square.
• Approximation provides a practical way to express irrationals in a usable form, especially when precision is less critical.

By rounding or using a calculator, you get an approximate value that can be easily interpreted and applied. This is very useful in calculations where exact values of irrational numbers aren't necessary.