PLEASE HELPPPPPWhich Of The Following Are Square Roots To The Number Below? Check All That Apply. 144

PLEASE HELPPPPPWhich Of The Following Are Square Roots To The Number Below? Check All That Apply. 144

Understanding square roots is a fundamental concept in mathematics that often confuses students and learners alike. When asked, "Which of the following are square roots of 144? Check all that apply," it’s essential to understand what a square root is and how to identify the correct values. In this article, we will explore the concept of square roots, specifically focusing on the number 144, and help you determine all the numbers that are square roots of 144. By the end, you will have a clear understanding of how to identify square roots and be able to confidently select all applicable options.

What Is a Square Root?

Definition of a Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. Mathematically, if x is a square root of n, then:

\[ x^2 = n \]

For example, 3 is a square root of 9 because:

\[ 3^2 = 9 \]

Similarly, -3 is also a square root of 9 because:

\[ (-3)^2 = 9 \]

It’s important to note that every positive real number has two square roots: a positive and a negative one.

Principal Square Root

The principal square root is the non-negative root of a number. When referring to "the square root" without any qualifiers, mathematicians typically mean the principal (positive) root. For 144, this is 12 because:

\[ 12^2 = 144 \]

However, when the question asks for all square roots, both positive and negative roots should be considered.

Square Roots of 144

Calculating the Square Roots of 144

To find the square roots of 144, ask: "What number, when multiplied by itself, results in 144?"

Let’s examine some options:

• 12:

\[ 12 \times 12 = 144 \]

• -12:

\[ -12 \times -12 = 144 \]

• Other possibilities?

Since 144 is a perfect square, its square roots are limited to the positive and negative roots of 12.

List of All Square Roots of 144

Based on the definition, the square roots of 144 are:

• Positive root: 12


• Negative root: -12

Hence, the numbers that are square roots of 144 are 12 and -12.

Checking Multiple-Choice Options

Suppose you are given multiple options and asked to select all the numbers that are square roots of 144. The options might be:

• 12


• -12


• 144


• -144


• 6


• -6

Let’s analyze each:

1. 12:

\[ 12^2 = 144 \quad \checkmark \]

1. -12:

\[ (-12)^2 = 144 \quad \checkmark \]

1. 144:

\[ 144^2 = 20,736 \neq 144 \quad \text{No} \]

1. -144:

\[ (-144)^2 = 20,736 \neq 144 \quad \text{No} \]

1. 6:

\[ 6^2 = 36 \neq 144 \quad \text{No} \]

1. -6:

\[ (-6)^2 = 36 \neq 144 \quad \text{No} \]

Correct options are 12 and -12.

Understanding Perfect Squares and Their Roots

What Is a Perfect Square?

A perfect square is a number that is the square of an integer. Since 144 is a perfect square (because 12×12 = 144), it has integer square roots.

Why Are Only 12 and -12 Roots of 144?

Because:

\[ 12 \times 12 = 144 \]
\[ (-12) \times (-12) = 144 \]

No other integers, when squared, produce 144, making these the only roots.

How to Find Square Roots of a Number

Step-by-Step Method

1. Factor the number: Find prime factorization if needed.
2. Identify perfect squares: Check if the number is a perfect square.
3. Find the positive root: Take the square root of the number.
4. Determine the negative root: Add the negative counterpart.
5. Verify: Multiply the potential roots by themselves to check if they produce the original number.

Example: Square Roots of 144

• Prime factorization:

\[ 144 = 2^4 \times 3^2 \]

• Square root:

\[ \sqrt{144} = \sqrt{2^4 \times 3^2} = 2^{2} \times 3^{1} = 4 \times 3 = 12 \]

• Roots are 12 and -12.

Common Mistakes to Avoid

• Confusing square roots with square: Remember, squaring a number multiplies it by itself, while taking a square root finds the original number that, when squared, produces the given number.
• Forgetting the negative root: Both positive and negative roots are valid unless specified otherwise.
• Assuming non-perfect squares have integer roots: Many numbers, like 50 or 200, do not have integer square roots.

Summary: Which Numbers Are Square Roots of 144?

In conclusion, the square roots of 144 are:

• 12 (positive root)


• -12 (negative root)

Any options other than these are not square roots of 144.

Additional Practice Questions

• Question 1: Is 144 a perfect square?

Answer: Yes, because 12×12 = 144.

• Question 2: What are the square roots of 81?

Answer: 9 and -9.

• Question 3: Find all square roots of 225.

Answer: 15 and -15.

Final Tips for Mastering Square Roots

• Always verify if the number is a perfect square before attempting to find square roots.
• Remember to consider both positive and negative roots unless specified.
• Use prime factorization for larger or more complex numbers to simplify finding roots.
• Practice with different numbers to build confidence in identifying square roots quickly.

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By understanding the concept of square roots and practicing with various numbers, you'll become proficient at solving these types of questions. For the specific case of 144, the key takeaway is that the square roots are 12 and -12.