Problem 24 - Algebra A Combined Function

Question

Use a calculator to approximate each square root to three decimal places. Check to see that each approximation is reasonable. $$ \sqrt{300} $$

Step-by-Step Solution

Answer

The square root of 300 is approximately 17.321.

Step 1: Identify the Nearest Perfect Squares

Before using the calculator, identify two perfect squares between which 300 falls. Notice that \(17^2 = 289\) and \(18^2 = 324\). Therefore, \(\sqrt{300}\) is between 17 and 18.

Step 2: Use the Calculator to Find the Approximation

Enter \(\sqrt{300}\) into the calculator. The calculator provides an approximation of \(\sqrt{300} \approx 17.321356\). Round this to three decimal places: 17.321.

Step 3: Validate the Approximation

Check the reasonableness of the approximation by squaring it. Compute \((17.321)^2 \approx 299.946041\). Since 299.95 is very close to 300, the approximation is reasonable.

Key Concepts

Understanding Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, the number 16 is a perfect square because it can be written as \(4 \times 4\) or \(4^2\).
Recognizing perfect squares is crucial when estimating square roots, especially when calculating without a calculator. Knowing the perfect squares helps to quickly narrow down the range in which a non-square number's root lies.
For instance, when you're tasked with estimating the square root of 300, first identify perfect squares around 300. We know that \(17^2 = 289\) and \(18^2 = 324\), indicating that \(\sqrt{300}\) falls between 17 and 18.
This is a fundamental step before diving into calculator usage because it sets a realistic expectation for the square root, allowing better estimation and cross-verification of results.

Calculator Approximations for Square Roots

Calculators are powerful tools for approximating square roots to a desired degree of precision. Here's how they make the task easier:

• First, input the number for which you need the square root, in this case, 300, into the calculator.
• The device does complex calculations internally to spit out a number that is a close approximation.

For \(\sqrt{300}\), the calculator gives us approximately 17.321356. This result is more accurate than what we could achieve manually without extensive effort.
One of the benefits of using a calculator is that it allows users to focus on understanding the reasonableness of the result rather than the nitty-gritty of the calculations. Ensure the calculated approximation falls within the expected range determined by the nearby perfect squares.
In this case, 17.321 fits between 17 and 18, thus validating the result. Using technology saves time and increases accuracy, two key perks when working with square roots.

Rounding to Decimal Places

Rounding numbers is an essential mathematical skill, particularly when handling approximations. After finding a square root using a calculator, the resulting number might contain many decimal places.
To simplify and standardize results, you'll often be asked to round to a certain number of decimal places. Let's break down how to round decimals effectively:

• Locate the decimal place to which you need to round, say the third decimal place.
• Check the number immediately after this decimal place. If it's 5 or greater, round up the last place. If it's less than 5, keep it as is.

Taking \(\sqrt{300}\approx 17.321356\), we round it to 17.321 because the fourth decimal digit (3) is less than 5.
Rounding makes numbers easier to work with while preserving a reasonable degree of accuracy, making calculations more straightforward without significantly compromising precision. It's a handy tool not only in mathematical exercises but also in real-world applications where exact numbers are unnecessary.