The Monomial $36x^4$ Is A Perfect Square. What Is The Square Root Of $36x^4$?A. $6x^2$ B. $6x^4$ C. $18x^2$ D. $18x^4$

Introduction

In algebra, a perfect square is a monomial or polynomial that can be expressed as the square of another monomial or polynomial. In this article, we will explore the concept of perfect squares and how to find the square root of a given monomial. We will use the monomial $36x^4$ as an example to demonstrate the process.

What is a Perfect Square?

A perfect square is a monomial or polynomial that can be expressed as the square of another monomial or polynomial. For example, the monomial $x^2$ is a perfect square because it can be expressed as $(x)^2$. Similarly, the monomial $x^4$ is also a perfect square because it can be expressed as $(x^2)^2$.

The Monomial $36x^4$ is a Perfect Square

The monomial $36x^4$ is a perfect square because it can be expressed as the square of another monomial. To find the square root of $36x^4$, we need to find the monomial that, when squared, gives us $36x^4$.

Breaking Down the Monomial

To find the square root of $36x^4$, we need to break down the monomial into its prime factors. The prime factorization of $36$ is $2^2 \cdot 3^2$, and the prime factorization of $x^4$ is $(x^2)^2$. Therefore, the prime factorization of $36x^4$ is $2^2 \cdot 3^2 \cdot (x^2)^2$.

Finding the Square Root

Now that we have the prime factorization of $36x^4$, we can find the square root by taking the square root of each factor. The square root of $2^2$ is $2$, the square root of $3^2$ is $3$, and the square root of $(x^2)^2$ is $x^2$. Therefore, the square root of $36x^4$ is $2 \cdot 3 \cdot x^2 = 6x^2$.

Conclusion

In conclusion, the monomial $36x^4$ is a perfect square, and its square root is $6x^2$. We found the square root by breaking down the monomial into its prime factors and taking the square root of each factor.

Answer

The correct answer is A. $6x^2$.

Why is the other options incorrect?

Option B, $6x^4$, is incorrect because it is not the square root of $36x^4$. The square root of $36x^4$ is $6x^2$, not $6x^4$.

Option C, $18x^2$, is incorrect because it is not the square root of $36x^4$. The square root of $36x^4$ is $6x^2$, not $18x^2$.

Option D, $18x^4$, is incorrect because it is not the square root of $36x^4$. The square root of $36x^4$ is $6x^2$, not $18x^4$.

Final Thoughts

Q: What is a perfect square in algebra?

A: A perfect square is a monomial or polynomial that can be expressed as the square of another monomial or polynomial.

Q: How do you find the square root of a perfect square?

A: To find the square root of a perfect square, you need to break down the monomial into its prime factors and take the square root of each factor.

Q: What is the prime factorization of $36x^4$?

A: The prime factorization of $36x^4$ is $2^2 \cdot 3^2 \cdot (x^2)^2$.

Q: How do you find the square root of $36x^4$?

A: To find the square root of $36x^4$, you need to take the square root of each factor in its prime factorization. The square root of $2^2$ is $2$, the square root of $3^2$ is $3$, and the square root of $(x^2)^2$ is $x^2$. Therefore, the square root of $36x^4$ is $2 \cdot 3 \cdot x^2 = 6x^2$.

Q: Why is $6x^4$ not the square root of $36x^4$?

A: $6x^4$ is not the square root of $36x^4$ because it is not the result of taking the square root of each factor in the prime factorization of $36x^4$. The correct square root of $36x^4$ is $6x^2$, not $6x^4$.

Q: Why is $18x^2$ not the square root of $36x^4$?

A: $18x^2$ is not the square root of $36x^4$ because it is not the result of taking the square root of each factor in the prime factorization of $36x^4$. The correct square root of $36x^4$ is $6x^2$, not $18x^2$.

Q: Why is $18x^4$ not the square root of $36x^4$?

A: $18x^4$ is not the square root of $36x^4$ because it is not the result of taking the square root of each factor in the prime factorization of $36x^4$. The correct square root of $36x^4$ is $6x^2$, not $18x^4$.

Q: What is the correct answer for the square root of $36x^4$?

A: The correct answer for the square root of $36x^4$ is $6x^2$.

Q: Why is it important to understand perfect squares in algebra?

A: Understanding perfect squares in algebra is important because it helps you to simplify expressions and solve equations. Perfect squares are a fundamental concept in algebra, and being able to identify and work with them is essential for success in mathematics.

Q: How can I practice finding the square root of perfect squares?

A: You can practice finding the square root of perfect squares by working through examples and exercises in your algebra textbook or online resources. You can also try creating your own examples and challenging yourself to find the square root of different perfect squares.

Q: What are some common mistakes to avoid when finding the square root of perfect squares?

A: Some common mistakes to avoid when finding the square root of perfect squares include:

• Not breaking down the monomial into its prime factors
• Not taking the square root of each factor in the prime factorization
• Not simplifying the expression after finding the square root
• Not checking your answer to make sure it is correct

By avoiding these common mistakes, you can ensure that you are finding the correct square root of perfect squares and building a strong foundation in algebra.