Problem 107 - Algebra A Combined Function

Question

Do not use a calculator. \(\sqrt{160}\) is closest to a. 10 b. \(13 \quad\) c. 20 d. 40

Step-by-Step Solution

Answer

The closest option is b. 13.

Step 1: Identify Perfect Squares around 160

Find the perfect squares that are close to 160. Recognize that 144 is the square of 12 and 169 is the square of 13. Therefore, the number whose square root is closest to 160 lies between 12 and 13.

Step 2: Estimate Square Root Value

Since 160 is closer to 169 than it is to 144, the square root of 160 is closer to 13 than it is to 12.

Step 3: Compare to Given Options

From the options provided (10, 13, 20, 40), 13 is the value that is closest to our estimation of the square root of 160 being near 13.

Key Concepts

Perfect Squares

Perfect squares are numbers that are the result of a number multiplied by itself. For instance, the perfect square of 12 is 144, because 12 times 12 equals 144. Similarly, 13 multiplied by 13 gives 169, making 169 a perfect square as well.
Understanding perfect squares is crucial in estimating square roots, especially when calculators aren't allowed. By identifying the two perfect squares that surround a number, you can figure out the approximate square root of that number.
So, if we're dealing with finding the square root of 160, we note that it lies between the perfect squares 144 and 169. This tells us the square root must lie between the numbers whose squares are 144 and 169—specifically 12 and 13.

Mathematical Estimation

Mathematical estimation involves making a reasonable guess about a mathematical value. It's especially useful when exact calculations are unnecessary or when a calculator is not available.
For estimating the square root of 160, once we have identified perfect squares nearby, we determine how close 160 is to either perfect square. Since 160 is closer to 169 than 144, we can reasonably estimate that the square root is closer to 13 rather than 12.
This estimation skill helps you solve problems more quickly and efficiently. Practicing estimation can improve your ability to assess numerical values in everyday situations.

Educational Problem Solving

Educational problem-solving is the process of working through a problem to discover a solution. In mathematics, this often involves applying different concepts and strategies to find the best answer.
The problem of estimating the square root of 160 without a calculator is a perfect example. It requires understanding of both perfect squares and estimation to arrive at the correct answer without directly calculating it.
Learning to solve problems in this manner can be applied to many different fields, enhancing critical thinking and analytical skills. By practicing educational problem-solving, you learn to identify core issues, apply the right strategies, and independently find solutions.