Problem 21 - Elementary Algebra

Question

Estimate each square root between two consecutive whole numbers. $$ \sqrt{70} $$

Step-by-Step Solution

Answer

\(8 < \sqrt{70} < 9\)

Step 1 - Identify Perfect Squares

Firstly, identify the perfect squares that are closest to 70. These are squares of whole numbers. Find two consecutive perfect squares such that: \ \(a^2 < 70 < b^2\).

Step 2 - Find Square Roots of Perfect Squares

The perfect squares close to 70 are 64 and 81. Calculate their square roots: \ \(\sqrt{64} = 8\) \ \(\sqrt{81} = 9\)

Step 3 - Locate \(\sqrt{70}\) Between Two Consecutive Whole Numbers

Since 70 is between 64 and 81, it follows that: \ \(\sqrt{64} < \sqrt{70} < \sqrt{81}\). Thus, \ 8 < \sqrt{70} < 9.

Key Concepts

Square Roots

A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 multiplied by 5 equals 25. Square roots can be challenging, but they become easier if you know some perfect squares and how to estimate between them. Remember, every positive number has two square roots: a positive and a negative. For instance, both 5 and -5 are square roots of 25. In daily math problems, we usually use the positive square root.

Perfect Squares

Perfect squares are numbers that are the result of an integer multiplied by itself. For example, 1, 4, 9, 16, 25, and 36 are perfect squares because:

• 1 = 1 * 1
• 4 = 2 * 2
• 9 = 3 * 3
• 16 = 4 * 4
• 25 = 5 * 5
• 36 = 6 * 6

Knowing perfect squares can help you estimate square roots. To estimate \(\root{70}\), find the perfect squares closest to 70. In this case, 64 (8^2) and 81 (9^2) are nearest. This tells us that \(\root{70}\) lies between 8 and 9.

Consecutive Whole Numbers

Consecutive whole numbers are numbers that follow each other in order without gaps. Examples are 1, 2, 3, or 8, 9, 10. When estimating square roots, you often need to find which two consecutive whole numbers the square root lies between. For \(\root{70}\), the perfect squares 64 and 81 help you identify this range. Since 64 (8^2) < 70 < 81 (9^2), you know \(\root{70}\) falls between 8 and 9. This approach can be used for other square root estimations too. Identify the perfect squares closest to your number, then find the consecutive whole numbers in between.