Problem 21 - Prealgebra

Question

Use a calculator to find each square root to the nearest tenth. $$\pm \sqrt{0.75}$$

Step-by-Step Solution

Answer

The square roots are approximately \(+0.9\) and \(-0.9\).

Step 1: Identify the Mathematical Operation

The problem requires finding the square root of the number 0.75. This involves determining a number which, when multiplied by itself, results in 0.75.

Step 2: Use a Calculator to Determine the Square Root

Enter 0.75 into a calculator and use the square root function to find its square root. The result is approximately 0.8660254.

Step 3: Round to the Nearest Tenth

Look at the number 0.8660254. Since the tenths place is 8 and it has a digit 6 in the hundredths place, round up to 0.9 as the nearest tenth.

Step 4: Consider the Positive and Negative Roots

The original problem asked for \(+\) and \(-\) square roots, meaning both \(\pm\sqrt{0.75}\). Thus, our answer includes both positive and negative roots: \(+0.9\) and \(-0.9\).

Key Concepts

Rounding Numbers

Rounding numbers is a fundamental math skill that helps simplify complex figures. When rounding to the nearest tenth, focus on the tenths and hundredths places.

• The tenths place is the first digit after the decimal point.
• The hundredths place is the second digit after the decimal point.

To round a number:

• If the hundredths digit is 5 or more, increase the tenths digit by one and drop all remaining digits.
• If the hundredths digit is less than 5, keep the tenths digit the same and drop all remaining digits.

In our case, 0.8660254 rounds to 0.9 because the tenths digit is 8 and the hundredths digit is 6, which is greater than 5. Hence, the tenths digit increases to 9.

Use of Calculators

Calculators are valuable tools for performing arithmetic operations quickly and accurately. In finding a square root, a calculator saves time compared to manual calculations, especially when dealing with non-perfect squares like 0.75.To use a calculator to find the square root:

• Enter the number into the calculator.
• Press the square root function key, often depicted as \( \sqrt{ } \).
• Read the displayed result, which is often to several decimal places.

A calculator simplifies obtaining results such as the square root of 0.75, calculated as approximately 0.8660254. Without a calculator, this process would require labor-intensive manual calculations. Hence, using a calculator ensures you reach accurate results efficiently, making it indispensable for both simple and complex mathematical tasks.

Positive and Negative Roots

Square roots can have both positive and negative values due to the nature of squaring a number. When you square a positive number, the result is positive. Squaring a negative number also results in a positive because the product of two negative numbers is positive. This means:

• The square root of any positive number, like 0.75, will have two solutions: a positive root and a negative root.
• These are represented as \( +\sqrt{} \) and \( -\sqrt{} \) in mathematical terms.

In our problem, both \( +\sqrt{0.75} = 0.9 \) and \( -\sqrt{0.75} = -0.9 \) are valid results. This reflects the dual nature of square roots, providing broader context for how squares relate to their roots. It showcases the importance of acknowledging both roots in math problems that involve finding square roots in equations and real-world applications.