Problem 80 - Prealgebra

Question

Use a calculator to approximate the square root to the nearest tenth. \(\sqrt{162}\)

Step-by-Step Solution

Answer

The square root of 162 to the nearest tenth is 12.7.

Step 1: Understand the Problem

The goal is to find the square root of 162 and round it to the nearest tenth. This means we need a decimal approximation of \(\sqrt{162}\).

Step 2: Use the Calculator

Input 162 into the calculator and use the square root function to find \(\sqrt{162}\). Write down the full decimal number that the calculator displays.

Step 3: Analyze the Result

The result from the calculator should be approximately 12.727922. In this step, we identify the digits after the decimal point to determine how to round it to the nearest tenth.

Step 4: Round to the Nearest Tenth

Look at the first digit after the decimal point, which is 7, and the next digit, which is 2. Since 2 is less than 5, we keep the first digit as 7, making the approximation 12.7.

Key Concepts

Using a Calculator

Using a calculator to find the square root of a number is extremely helpful, especially for numbers that don't have a neat or obvious square root. Here’s a simple guide to using a calculator for square roots:

• Most scientific and smartphone calculators have a square root button, often labeled as \( \sqrt{} \).
• Simply enter the number whose square root you wish to find, in this case, 162.
• Press the square root button, and the calculator will display the result, which is a decimal approximation of the true square root.

In our example, entering 162 and pressing the square root button will show a result like 12.727922. This provides us with a detailed decimal, which is what we need to begin the process of rounding.

Rounding Decimals

Rounding decimals is an essential skill, particularly when dealing with long decimal numbers from calculations like square roots. Let’s break down the process: - First, identify the place value to which you need to round. For rounding to the nearest tenth, focus on the first decimal place. - Observe the digit in the place right after the one you are rounding to. This digit determines whether you round up or stay. For the number 12.727922: - The digit in the tenths place is 7. - The digit in the hundredths place is 2. - Since the hundredths digit (2) is less than 5, the tenths digit (7) remains unchanged. Thus, rounding 12.727922 to the nearest tenth gives us 12.7.

Square Root Calculation

Understanding square root calculation helps us know why using a calculator is necessary for certain numbers. A square root is a number that, when multiplied by itself, gives the original number. For example:- The square root of 25 is 5 because \( 5 \times 5 = 25 \).- The square root symbol \( \sqrt{} \) represents this operation.Not all numbers have an easy integer square root. Like 162, many numbers result in decimals when we calculate their square roots.The calculation involves finding such a number whose square squares back to our original number. This complexity is why calculators are useful; they handle the intricacies and give us precise decimals quickly. Remember that these decimals are approximations, which makes rounding them necessary for practical use.