Problem 36 - Intermediate Algebra

Question

Use a calculator to find each square root. Give each answer to four decimal places. See Objective 1. $$ \sqrt{340} $$

Step-by-Step Solution

Answer

The square root of 340 to four decimal places is 18.4391.

Step 1: Understand the Problem

We need to find the square root of 340 and provide the answer to four decimal places. The square root operation asks what number multiplied by itself gives the result 340.

Step 2: Use a Calculator

Input the number 340 into a calculator and apply the square root function. Most calculators have a square root button, usually labeled as "√x".

Step 3: Record the Calculator Result

The calculator displays the result as approximately 18.4390889146. This number is the square root of 340.

Step 4: Round to Four Decimal Places

Round 18.4390889146 to four decimal places. To do this, look at the fifth decimal place, which is 8. Since it's 5 or greater, round the fourth decimal place up from 9 to 10, but this carries over and changes the third decimal place, ending the rounding at 18.4391.

Key Concepts

Rounding Decimals

Rounding decimals is a fundamental concept in numerical analysis. It involves approximating a number to a certain degree of accuracy or decimal places.
Understanding how to round numbers correctly is crucial for many mathematical calculations.

• Identify the number of decimal places to round to. In our example, we need to round to four decimal places.
• Examine the digit immediately after the desired decimal place (the fifth decimal place). If this digit is 5 or higher, increase the last kept digit by one.
• If the digit is less than 5, keep the last kept digit unchanged.

In the \( \sqrt{340} \) example, the calculated square root was approximately 18.4390889146. To round this to four decimal places:
- Look at the fifth decimal digit: it's 8, which is greater than 5.
- Therefore, we round the fourth decimal place up, resulting in 18.4391.

Square Root Operation

The square root operation is a mathematical function that identifies a number which, when multiplied by itself, yields the original number under the square root. For example, the square root of 340 is found as follows:

• Visualize it as \( \sqrt{340} \).
• This asks you to find a number \( x \), such that \( x \times x = 340 \).
• Use a calculator to perform the operation efficiently.

Remember that square roots can be both positive and negative because both \( x \) and \( -x \) satisfy the equation \( x^2 = y \). However, in most cases, especially with calculators, we use only the principal (positive) square root.
Thus, when you calculate \( \sqrt{340} \), you are usually interested in the principal value.

Using Calculators for Math Problems

Calculators are invaluable tools for solving mathematical problems, especially when dealing with complex calculations or ensuring precision. They handle the square root operation using a simple keypress, making it easy to find answers quickly and accurately.
Here's how to employ a calculator effectively:

• Enter the number you want the square root of (e.g., 340).
• Locate and press the square root function, often labeled as \( \sqrt{} \) or "√".
• View the result. The calculator provides an approximate value, often to more decimal places than required.

When using calculators, be mindful of the level of precision you need. For your exercise, rounding to four decimal places was specified. Hence, always ensure you adjust your answer according to the exercise's requirements.
This not only enhances accuracy but also ensures alignment with educational standards.