Problem 7 - Fundamentals of Mathematics

Question

Find the following roots using only a knowledge of multiplication. $$\sqrt{64}$$

Step-by-Step Solution

Answer

The square root of 64 is 8.

Step 1: Understand the Problem

We need to find the square root of the number 64, using our understanding of multiplication. The square root of a number is another number which, when multiplied by itself, gives the original number.

Step 2: Recall Known Squares

Let's recall some small square numbers: - The square of 1 is 1 (since 1 × 1 = 1) - The square of 2 is 4 (since 2 × 2 = 4) - The square of 3 is 9 (since 3 × 3 = 9) Continue listing these until you find the one that matches 64.

Step 3: Identify the Matching Square

List more squares: - The square of 4 is 16 (since 4 × 4 = 16) - The square of 5 is 25 (since 5 × 5 = 25) - The square of 6 is 36 (since 6 × 6 = 36) - The square of 7 is 49 (since 7 × 7 = 49) - The square of 8 is 64 (since 8 × 8 = 64) We found that 8 multiplied by itself gives 64.

Step 4: Conclude the Solution

Since 8 multiplied by 8 gives us 64, the square root of 64 is 8.

Key Concepts

Multiplication

Multiplication is a basic arithmetic operation that involves combining groups of numbers. It's often referred to as repeated addition. For example, when you multiply 4 by 3, you are essentially adding four groups of 3 together: 3 + 3 + 3 + 3, which equals 12.

• It's represented by symbols like '×', '*', or sometimes with parentheses.
• This operation is fundamental in finding square roots because square roots involve reversal of this concept.

Understanding multiplication helps us identify the root of a number by recognizing which number, when multiplied by itself, results in another number. In our problem, we used multiplication to find that 8 times 8 equaled 64, thus identifying the square root of 64 as 8.

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. They are the result of squaring whole numbers. For example, 1, 4, 9, 16, and 64 are perfect squares because they are 1×1, 2×2, 3×3, 4×4, and 8×8 respectively.

• Perfect squares are important in mathematics because they simplify the process of finding square roots.
• By memorizing small perfect squares, students can more easily identify squares of larger numbers.

If you recognize that a number is a perfect square, you can immediately determine its square root without having to multiply different numbers. In our exercise, knowing 64 is a perfect square allowed us to quickly find its square root, which is 8.

Arithmetic

Arithmetic is the branch of mathematics dealing with numbers and the basic operations: addition, subtraction, multiplication, and division. It provides the foundational rules and operations used in all math problems. In the context of our exercise, arithmetic allowed us to systematically determine the square root of 64.

• Knowing arithmetic operations allows us to break down complex problems into simpler steps.
• It was through understanding arithmetic principles that we could apply multiplication to find the square root of 64.

Arithmetic helps us not just in math, but also in understanding how numbers relate to each other in various real-world applications. With a firm grasp of arithmetic, other math concepts like square roots can become much more approachable.