Problem 25 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Check the results by squaring each root. $$ \sqrt{144} $$

Step-by-Step Solution

Answer

The square root of 144 is 12. When 12 is squared, it results in 144. This confirms that the square root of 144 is correctly calculated as 12.

Step 1: Calculate Square Root

Start by finding the square root of 144. The square root operation is symbolized by \(\sqrt{} \), which is asking what number, when multiplied by itself, gives the number under the root. One can often find the square root of a perfect square number (like 144) by recalling multiplication facts.

Step 2: Square the Root

The next step is to check the answer by squaring the root. The process of squaring a number is multiplying that number by itself.

Step 3: Compare Results

The final step is to compare the squared result to the original number under the square root. If they match, the square root was calculated correctly.

Key Concepts

Understanding Perfect Squares

When you hear "perfect square," it refers to a number that is the product of another integer multiplied by itself. For example, 144 is a perfect square because it equals 12 times 12. Recognizing perfect squares is important, as they simplify the process of finding square roots.

• Common perfect squares: 1, 4, 9, 16, 25, 36, and so on.
• These numbers are easy to work with since their square roots are whole numbers.
• Learning to recognize them can save time when evaluating expressions involving square roots.

By identifying 144 as a perfect square, you quickly see that its square root is 12 because 12 multiplied by itself returns to the original number of 144.

The Process of Squaring a Number

Squaring a number is essentially multiplying the number by itself. This operation plays a significant role in reversing the square root operation and verifying calculations.

• For example, the square of 12, as in our original exercise, is calculated as follows:
• \[12 \times 12 = 144\]This shows that squaring the root of 144 confirms that 12 is correct.
• Squaring helps build understanding of multiplication concepts, reinforcing number relationships.

Whenever you encounter square roots in math, it's a good habit to square the result to confirm correctness.

Evaluating Expressions Involving Square Roots

Evaluating expressions with square roots involves finding the value of the expression by determining the root. It also includes verifying if the calculation was correct through the process of squaring.

• Start by identifying if the number under the root is a perfect square.
• Compute the square root using multiplication facts or mental math.
• Ensure accuracy by squaring your result to see if it matches the initial number.
• In the case of complex expressions, handle each root separately and simplify systematically.

Evaluating expressions involving square roots requires patience and understanding of the properties of numbers, which helps in mastering mathematical proficiency.