Problem 38 - Algebra 2

Question

Use a calculator to approximate each value to three decimal places. $$ -\sqrt{147} $$

Step-by-Step Solution

Answer

The approximate value is -12.124.

Step 1: Identify the Expression

The given expression to approximate using a calculator is \(-\sqrt{147}\). This indicates that we need to find the negative square root of 147.

Step 2: Calculate the Square Root

Using a calculator, find the square root of 147. The square root of 147 is approximately 12.124.

Step 3: Apply the Negative Sign

Apply the negative sign to the square root result. Since we need to approximate \(-\sqrt{147}\), take the value 12.124 obtained in the previous step and multiply by -1 to get -12.124.

Key Concepts

Calculator Use

Using a calculator properly can make difficult math problems much easier. In exercises involving square roots, such as finding \( -\sqrt{147} \, \), a calculator is extremely helpful for quick and precise calculations. Here's how you can make the most of calculator use when approximating square roots:

• Ensure your calculator has a square root function, often indicated by a symbol that looks like \( \sqrt{} \).
• Input the number inside the square root function (147 in this case), and press the square root button. What you get is the positive square root, 12.124.
• Be mindful of the number of decimal places required. Rounding is crucial, especially if your calculator provides more digits than needed.

Calculators provide fast approximations, but always double-check your entries to avoid mistakes.

Negative Numbers

Understanding how negative numbers work is key when dealing with expressions like \( -\sqrt{147} \). A negative number, simply put, is any number less than zero. These numbers are often represented with a negative sign (minus) in front of them.

In the context of square roots, the negative sign in front of \( \sqrt{147} \) indicates that the result is the negative form of the square root. In simpler terms, once you've found the square root of 147 using your calculator (which is 12.124), you must attach a negative sign to depict the final value as -12.124.

While working with negative numbers, remember that:

• A negative times a positive equals a negative (-1 x 147 = -147).
• The square root of a number is always positive. However, applying a negative sign in front makes it negative, as seen here.

Always be careful with negative signs so you don't misrepresent your final answers.

Approximation

Approximating numbers helps when an exact value isn't necessary or the number is too complex for simple calculations. When approximating square roots like \( -\sqrt{147} \), it's important to understand when and how to round appropriately.

Approxiating means providing a value that is close enough to the real number by simplifying it to a certain number of decimal places. For this exercise, we rounded to three decimal places. This increases the usability of the approximated number for further calculations or real-world applications:

• First, calculate the square root precisely with a calculator.
• Next, determine how many decimal places are needed—in this case, three.
• Use normal rounding rules to adjust the number to this precision.

For example, the calculator might give 12.12345 as a result. To three decimals, this number becomes 12.124. Approximating helps to represent complex numbers in a more manageable form, particularly in routine calculations.