Question

The square root of 73 lies between which two integers? (Note: You may not use a calculator.) (A) 6 and 7 (B) 7 and 8 (C) 8 and 9 (D) 9 and 10

Short answer

The square root of 73 lies between the integers 8 and 9. Therefore, the correct answer is (C) 8 and 9.

Step-by-step solution

Find the square of integers until surpassing 73

Start with 1 and find the square of each increasing integer until the result surpasses 73. For example, \(1^2=1\), \(2^2=4\), \(3^2=9\), and so on.

Identify the two numbers

Identify the two integers where one is the highest integer with a square less than 73 and one is the smallest integer with a square more than 73.

Confirm the answer by checking the options

Check the answer obtained above with the given options. Following these steps: In Step 1, \(6^2 = 36\), \(7^2 = 49\), \(8^2 = 64\) and \(9^2 = 81\). In Step 2, it can be noticed that 73 falls between the squares of 8 and 9. In Step 3, confirming this with the answer options, the correct answer is C) 8 and 9 as this aligns with the results from step 2.

Key concepts

Estimating Square Roots

Estimating square roots is an essential skill, particularly when working without the aid of a calculator. It involves understanding that square roots are numbers which, when multiplied by themselves, yield a specific perfect square. To estimate the square root of a non-perfect square like 73, you can find the closest perfect squares before and after it.

Here's how you can do it: look for perfect squares of whole numbers that are close to 73. Since you know that the perfect squares of 8 and 9 are 64 and 81, respectively, you can conclude that the square root of 73 must fall between these two numbers. Thus, without the need for precise calculation, you can estimate that the square root of 73 falls between 8 and 9.

Non-Calculator Math

Handling math problems without a calculator may seem daunting, but it sharpens your number sense and mental math skills. When you can't reach for a calculator, you rely on understanding mathematical concepts, such as the relationship between numbers and their squares. In our exercise, by knowing that 8 squared is 64 and 9 squared is 81, you leverage this knowledge to estimate square roots. Non-calculator math encourages the learning of efficient methods and shortcuts, which not only aid in mental calculation but also enhance problem-solving abilities in various mathematical scenarios.

Square Numbers

Square numbers, or perfect squares, are the product of any integer multiplied by itself. They form the foundation for understanding square roots. Familiarity with the sequence of square numbers (1, 4, 9, 16, ...) is crucial since it aids in recognizing patterns and quickly identifying estimates for square roots. Square numbers grow at an increasing rate, so the difference between consecutive squares gets larger as you move up the number line. Understanding these properties can help you swiftly find which two consecutive square numbers a given number falls between, as done with the square root estimation for 73.

GED Math Practice

The General Educational Development (GED) test assesses knowledge in various subjects, including math. GED math practice involves mastering concepts without the use of calculators for some parts. Exercises like estimating square roots not only prepare you for such test sections but also build conceptual understanding and efficient problem-solving skills. It's important to practice various math techniques, like approximation and number comparison, to improve your speed and accuracy for the GED exam. Working through multiple problems, such as the given square root estimation, is excellent preparation for the test's non-calculator segment.