Problem 8 - Fundamentals of Mathematics

Question

Find the following roots using only a knowledge of multiplication. $$\sqrt{100}$$

Step-by-Step Solution

Answer

10

Step 1: Understand the Problem

We need to find the square root of 100, which means identifying a number that, when multiplied by itself, results in 100.

Step 2: Recall Perfect Squares

Recall the list of perfect squares to find the closest number whose square is 100. For example, some perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, etc.

Step 3: Identify the Perfect Square

From the list, notice that 10 is a number whose square is 100, because 10 × 10 = 100.

Step 4: Verify the Calculation

Verify by multiplying to ensure accuracy: 10 times 10 equals 100, so 10 is indeed the correct square root of 100.

Key Concepts

Understanding Perfect Squares

A perfect square is a number that can be expressed as the product of an integer multiplied by itself. For instance, 100 is a perfect square because it can be written as 10 multiplied by 10. Recognizing perfect squares is an essential skill in mathematics because it allows us to quickly determine square roots.

Here's a quick list of some common perfect squares:

• 1 (1 × 1)
• 4 (2 × 2)
• 9 (3 × 3)
• 16 (4 × 4)
• 25 (5 × 5)
• 36 (6 × 6)
• 49 (7 × 7)
• 64 (8 × 8)
• 81 (9 × 9)
• 100 (10 × 10)

When solving square root problems, this list can be very helpful, particularly if you remember the numbers as benchmarks. Knowing perfect squares makes it simple to handle square roots without complex calculations.

The Role of Multiplication in Finding Square Roots

Multiplication is the key operation to use when finding square roots manually, especially if you're looking for a number whose square gives a particular value. To find the square root of a perfect square, you need to consider multiplication in reverse.

For example, if you need to find \(\sqrt{100}\), think about which number multiplied by itself gives 100. This requires:

• Knowing your multiplication facts
• Recognizing patterns in numbers

If you're working with small numbers, checking multiplication facts can be done quickly, like realizing that 10 × 10 = 100, so the square root of 100 is 10.

Mathematical Verification for Accuracy

Verification ensures that your mathematical solution is accurate and correct. In the context of finding square roots, verification involves taking the number you believe to be the square root and squaring it to check if it gets you back to the original number.

To verify \(\sqrt{100} = 10\), you multiply 10 by itself:

• 10 × 10 = 100

Since the multiplication results in 100, you confirm that 10 is indeed the correct square root. Verification is an excellent habit to develop because it helps catch mistakes and misunderstandings before finalizing your answer.