Problem 65 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt{144}\)

Step-by-Step Solution

Answer

The square root of 144 is 12.

Step 1: Understand the Square Root Concept

The square root of a number is a value that, when multiplied by itself, gives the original number. We need to find a number that, when multiplied by itself, equals 144.

Step 2: Identify Potential Candidate Numbers

Consider numbers that could be multiplied by themselves to give 144, particularly integers. We suspect that it is between 10 and 15, because 10 squared is 100 and 15 squared is 225.

Step 3: Test Potential Candidates

Start with a middle number in our range, such as 12. Calculate \(12 \times 12 = 144\). Since the product is 144, 12 is indeed the square root of 144.

Step 4: Verify Using a Calculator

Use a calculator to check the calculation: input \(12 \times 12\) to verify it equals 144.

Step 5: Confirm the Solution

Since we verified that 12 squared is 144, the square root of 144 is confirmed as 12.

Key Concepts

Multiplication

In mathematics, multiplication is a basic operation that involves combining equal groups to find the total number of items. When we talk about multiplication in the context of finding square roots, like in our problem with \( \sqrt{144} \), we're looking for a specific number that, when multiplied by itself, results in 144. This involves an understanding of both multiplication and the nature of square numbers.
To solve \( \sqrt{144} \), consider potential candidates. This means identifying numbers that could multiply by themselves to equal 144. Some helpful tips for this process include:

• Consider the range of possibilities. With 144 being a perfect square, start with numbers you already know like the squares of 10, 11, 12, etc.
• Use known multiplication facts. This helps to ignore less likely candidates quickly by knowing that 10 squared is 100, and 15 squared is 225.

Multiplication allows us to be efficient by testing fewer numbers, leading to the discovery that 12 times itself equals 144, making 12 the square root. Understanding multiplication thus allows you to narrow down possibilities and verify answers confidently.

Integers

Integers are whole numbers that can be positive, negative, or zero. In the case of square roots like \( \sqrt{144} \), we are interested in finding which integer, when squared, gives the original number, 144.
Working with integers is particularly convenient when dealing with square roots because integer multiplication is straightforward and quickly provides clear, exact results. Here’s why integers are helpful:

• They simplify the calculation process. Integer operations are simpler compared to decimals and fractions, which can involve complex arithmetic.
• Perfect squares, such as 144, are often the squares of integers. Knowing some common perfect squares can speed up the process of finding square roots.

In our example, through a simple trial starting from easily manageable numbers like 11, 12, and 13, we quickly found that the integer 12, when multiplied by itself, equals 144. This discovery shines the importance of familiarity with simple integer operations in quickly solving mathematical problems.

Calculator Verification

Even after using multiplication to manually find the square root, it's always helpful to back up your result with a calculator verification. Calculators are incredibly reliable tools for ensuring accuracy in arithmetic operations. This adds a layer of confidence to your solution. Here's how verification with a calculator is beneficial in our problem of \( \sqrt{144} \):

• Efficiency: Quickly re-check calculations without the risk of manual error. Simply input \(12 \times 12\) and confirm the result is 144.
• Assurance: Provides a safety net that your manual operation on paper or mentally is correct. This is especially useful for complex numbers where mistakes are more likely.
• Learning: Helps reinforce your understanding when your result is confirmed, boosting self-assurance in mathematical skills.

By using a calculator to check your work, you made sure that the logical steps and multiplication used were correct, confirming that 12 is indeed the root of 144. This step consolidates both learning and accuracy.