Problem 15 - Intermediate Algebra

Question

Use a calculator to approximate each square root to 3 decimal places. $$ \sqrt{38} $$

Step-by-Step Solution

Answer

The approximate value of \( \sqrt{38} \) is 6.164.

Step 1: Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4 because 4 × 4 = 16. We are tasked with finding an approximation of \( \sqrt{38} \) to three decimal places.

Step 2: Using a Calculator

On your calculator, enter the number 38, then press the square root button (usually represented by \( \sqrt{x} \)). This will give you the result for \( \sqrt{38} \) more precisely.

Step 3: Rounding the Result

The calculator provides the square root as a long decimal. The aim is to round this value to three decimal places. Typically, the rule for rounding is to look at the fourth decimal place and round the third decimal place up if the fourth decimal is 5 or more.

Key Concepts

Calculator Usage

Using a calculator to approximate square roots is a handy skill that allows us to get accurate results quickly. When you are asked to approximate a square root, like \( \sqrt{38} \), a calculator becomes crucial.To use a calculator effectively, start by entering the number for which you need the square root, in this case, 38. On most calculators, there's a button labeled \( \sqrt{x} \) or similar, which you need to press after you've entered the number. This button performs the calculation for you, saving you from manually approximating by hand. Once you've pressed this button, the display will show a long decimal result which represents the square root of 38 with a very high degree of precision. Do remember to check your calculator manual if you encounter any trouble finding the square root function. Understanding how your calculator works is key to obtaining accurate mathematical results.

Rounding Decimals

After using your calculator to find \( \sqrt{38} \), you will see a long string of numbers. To keep things simple and neat, you have to round this number. Rounding decimals is about finding a simpler, approximate value that is easier to work with.The task here is to round to three decimal places. Start by counting three digits after the decimal point. Suppose the number is 6.164724. First, focus on the third decimal place, which is 4 in this scenario.

• Now look at the fourth number after the decimal, which is 7.
• If this number is 5 or more, you round the third digit up. If it's less than 5, you leave the third digit as is.
• Since 7 is greater than 5, we round 4 up to 5, changing the number to 6.165.

Rounding helps in managing numbers efficiently, especially when handling more complex calculations.

Understanding Square Roots

Square roots can initially be confusing, but they're just a way of working backward from multiplication. When you see \( \sqrt{38} \), it represents a number which, when squared (or multiplied by itself), results in 38.Imagine you know 6 squared is 36, a number close to 38. So, \( \sqrt{38} \) should be just a bit more than 6. This intuition can guide you when estimating before using a calculator or in mental math assessments.Although \( \sqrt{38} \) does not result in a neat whole number like \( \sqrt{36} \) does, it's you approximating these in-between numbers that will strengthen your understanding of math concepts. Overall, understanding square roots is about recognizing the relationship between numbers and enhancing your mathematical intuition.