Which Term Is A Perfect Square Of The Root $3x^4$?A. $6x^8$ B. $ 6 X 16 6x^{16} 6 X 16 [/tex] C. $9x^8$ D. $9x^{16}$

by ADMIN 128 views

Introduction

In mathematics, a perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as 4^2. Similarly, a perfect square of a root can be found by squaring the root. In this article, we will explore which term is a perfect square of the root 3x^4.

Understanding the Concept of Perfect Square

A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as 4^2. Similarly, a perfect square of a root can be found by squaring the root. To find the perfect square of a root, we need to square the root and simplify the expression.

Squaring the Root 3x^4

To find the perfect square of the root 3x^4, we need to square the root. We can do this by multiplying the root by itself.

3x4×3x4=(3x4)12×(3x4)12\sqrt{3x^4} \times \sqrt{3x^4} = (3x^4)^{\frac{1}{2}} \times (3x^4)^{\frac{1}{2}}

Using the property of exponents, we can simplify the expression as follows:

(3x4)12×(3x4)12=(3x4)12+12(3x^4)^{\frac{1}{2}} \times (3x^4)^{\frac{1}{2}} = (3x^4)^{\frac{1}{2} + \frac{1}{2}}

Simplifying further, we get:

(3x4)12+12=(3x4)1(3x^4)^{\frac{1}{2} + \frac{1}{2}} = (3x^4)^1

Simplifying the expression, we get:

(3x4)1=3x4(3x^4)^1 = 3x^4

However, we are looking for the perfect square of the root 3x^4, not the root itself. To find the perfect square, we need to square the root again.

(3x4)2=9x8(3x^4)^2 = 9x^8

Analyzing the Options

Now that we have found the perfect square of the root 3x^4, let's analyze the options.

A. $6x^8$

B. $6x^{16}$

C. $9x^8$

D. $9x^{16}$

Conclusion

Based on our analysis, we can conclude that the perfect square of the root 3x^4 is $9x^8$. This is because we squared the root 3x^4 to get 9x^8.

Answer

The correct answer is C. $9x^8$.

Final Thoughts

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as 4^2.

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

A: To find the perfect square of a root, you need to square the root and simplify the expression. For example, to find the perfect square of the root 3x^4, you would square the root as follows:

3x4×3x4=(3x4)12×(3x4)12\sqrt{3x^4} \times \sqrt{3x^4} = (3x^4)^{\frac{1}{2}} \times (3x^4)^{\frac{1}{2}}

Q: What is the property of exponents that I need to use to simplify the expression?

A: You need to use the property of exponents that states:

(am)n=amn(a^m)^n = a^{mn}

This property allows you to simplify the expression by multiplying the exponents.

Q: How do I simplify the expression using the property of exponents?

A: To simplify the expression, you need to multiply the exponents as follows:

(3x4)12×(3x4)12=(3x4)12+12(3x^4)^{\frac{1}{2}} \times (3x^4)^{\frac{1}{2}} = (3x^4)^{\frac{1}{2} + \frac{1}{2}}

Simplifying further, you get:

(3x4)12+12=(3x4)1(3x^4)^{\frac{1}{2} + \frac{1}{2}} = (3x^4)^1

Q: What is the final simplified expression?

A: The final simplified expression is:

(3x4)1=3x4(3x^4)^1 = 3x^4

However, we are looking for the perfect square of the root 3x^4, not the root itself. To find the perfect square, we need to square the root again.

Q: How do I square the root again?

A: To square the root again, you need to multiply the root by itself as follows:

(3x4)2=9x8(3x^4)^2 = 9x^8

Q: What is the perfect square of the root 3x^4?

A: The perfect square of the root 3x^4 is $9x^8$.

Q: What are some common mistakes to avoid when finding perfect squares?

A: Some common mistakes to avoid when finding perfect squares include:

  • Not squaring the root correctly
  • Not simplifying the expression correctly
  • Not using the property of exponents correctly

Q: How can I practice finding perfect squares?

A: You can practice finding perfect squares by working through examples and exercises. You can also use online resources and practice tests to help you prepare.

Q: What are some real-world applications of perfect squares?

A: Perfect squares have many real-world applications, including:

  • Algebra and geometry
  • Calculus and analysis
  • Physics and engineering
  • Computer science and programming

Q: How can I use perfect squares in my daily life?

A: You can use perfect squares in your daily life by applying mathematical concepts to real-world problems. For example, you can use perfect squares to calculate the area of a room or the volume of a container.

Conclusion

In this article, we have answered some frequently asked questions about perfect squares. We have covered topics such as what is a perfect square, how to find the perfect square of a root, and common mistakes to avoid. We hope that this article has provided you with a better understanding of perfect squares and how to use them in your daily life.