Problem 63 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt{64}\)

Step-by-Step Solution

Answer

The square root of 64 is 8.

Step 1: Identify the Operation

Recognize that the problem requires you to find the square root of 64, which means looking for a number that multiplies by itself to give 64.

Step 2: Guess and Check

Start by thinking of a number that squares to 64. Consider smaller numbers like 2, 3, 4 etc., until you find one that works.

Step 3: Choose a Candidate

Consider \(8\*8 = 64\). This suggests that 8 is a strong candidate for the square root of 64.

Step 4: Verify with Multiplication

Verify your solution by multiplying the candidate, 8, by itself to see if the product is 64. \(8\times8 = 64\) confirms this candidate.

Step 5: Check Using a Calculator

Use a calculator to find \(\sqrt{64}\) and verify that the answer is indeed 8.

Key Concepts

Square Root Calculation

Finding the square root of a number means determining which number, when multiplied by itself, yields the original number. For example, to find \( \sqrt{64} \), you're looking for a number that, when squared, equals 64. An easy approach to square root calculation is starting with smaller numbers. Testing \(2\): \(2 \times 2 = 4\). Obviously, too small. Trying \(3\): \(3 \times 3 = 9\). Still not 64. Then \(4 \): \(4 \times 4 = 16\). Keep going until you reach \(8 \times 8 = 64\). Bingo! You've found it.Break this down into steps for clarity:

• Start with smaller numbers.
• Multiply each candidate by itself.
• Stop when you find a match with the original number, 64 in this case.

This approach uses logical guesswork guided by mathematical operation order and multiplication.

Verification with Multiplication

Once you have a candidate for the square root, it’s crucial to verify it. Verification through multiplication ensures confidence in your answer.Take \(8\), our candidate for \(\sqrt{64}\), and check by calculating \(8 \times 8 \). When it equals 64, you're assured that 8 is the square root. Here’s the process:

• Take the candidate found through guesswork and calculation.
• Multiply it by itself (square it).
• If the result matches 64, congratulations, your candidate is correct.

This verification step confirms assumptions made during calculation and offers certainty.

Calculator Verification

Using a calculator is an excellent way to verify your square root calculation. Calculators provide a quick and accurate way to confirm your answers after manually finding a square root.After determining that \(8\) is your candidate for \( \sqrt{64} \) based on manual calculations:

• Input \( \sqrt{64} \) into your calculator.
• Check the displayed result.
• A reading showing \(8\) confirms the manual method's accuracy.

Calculator verification serves as a reliable tool, especially for complex numbers, enhancing confidence in your mathematical skills.