Problem 6 - Prealgebra

Question

Find each of the following square roots without using a calculator. $$\sqrt{144}$$

Step-by-Step Solution

Answer

The square root of 144 is 12.

Step 1: Understand the Problem

We need to find the square root of 144, which involves determining what number, when multiplied by itself, equals 144.

Step 2: Recall Perfect Squares

Recall that the perfect square numbers close to 144 are 121 (from 11, since \(11^2 = 121\)) and 169 (from 13, since \(13^2 = 169\)). This suggests that \(\sqrt{144}\) is likely 12.

Step 3: Verify by Squaring

To verify, calculate \(12^2\) and check if it equals 144. We know \(12 \times 12 = 144\), confirming that 144 is indeed the square of 12.

Step 4: Conclude the Solution

Since \(12^2 = 144\), the square root of 144 is 12. Thus, \(\sqrt{144} = 12\).

Key Concepts

Understanding Perfect Squares

Perfect squares are numbers that are the result of a whole number being multiplied by itself. For example, if you take the number 12 and multiply it by itself, you get 144. This means 144 is a perfect square because it is equal to 12 squared. Understanding perfect squares is foundational because they form the basis of finding square roots easily.
Knowing common perfect squares can be very useful. Here are some examples:

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

Recognizing these numbers quickly can help in mental math when determining square roots without a calculator. This memory trick simplifies the process and enhances your math skills.

Simplifying with Mental Math

Mental math involves doing arithmetic calculations without the help of any devices. It can sound intimidating, but with practice, it becomes second nature. For instance, calculating the square root of 144 mentally is straightforward if you remember your perfect squares.
When you encounter a problem like \(\sqrt{144}\), you think:

• "What number squared gives 144?"
• You recall that 12 times 12 yields 144.
• Therefore, the square root of 144 is 12.

Practicing mental math is a skill you can develop over time. Start by mastering smaller numbers and progressively move to larger ones. This way, when faced with a challenge like finding the square root, you can quickly arrive at the answer with confidence and ease.

Enhancing Through Mathematical Reasoning

Mathematical reasoning is a powerful tool that helps you solve problems methodically. When calculating \(\sqrt{144}\), reasoning can guide you to skip unnecessary steps, leading to faster and more efficient solutions.
This process involves:

• First understanding what the problem is asking. Here, it's asking for a number that multiplies by itself to give 144.
• Next, thinking about where 144 fits between known perfect squares, like 121 \(11^2\) and 169 \(13^2\).
• Then, reasoning that since 144 is between these values, the square root is almost certainly 12.
• Finally, checking this by squaring 12 to ensure \(12^2 = 144\).

By using logical steps and deduction, mathematical reasoning makes problems simpler and more approachable. This skill doesn't just apply to perfect squares or square roots but to all areas of math. Continuously practicing this thought process enhances overall problem-solving abilities.