Which Choice Is Equivalent To The Product Below When X ≥ 0 X \geq 0 X ≥ 0 ? 6 X 2 ⋅ 18 X 2 \sqrt{6 X^2} \cdot \sqrt{18 X^2} 6 X 2 ​ ⋅ 18 X 2 ​ A. 108 X 2 \sqrt{108 X^2} 108 X 2 ​ B. 6 X 2 3 6 X^2 \sqrt{3} 6 X 2 3 ​ C. 6 3 X 6 \sqrt{3 X} 6 3 X ​ D. 6 18 X 6 \sqrt{18 X} 6 18 X ​

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Introduction

Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the product of two square roots, specifically 6x2⋅18x2\sqrt{6 x^2} \cdot \sqrt{18 x^2}, and determine which choice is equivalent to this product when x≥0x \geq 0.

Understanding Square Roots

Before we dive into simplifying the given expression, let's quickly review the concept of square roots. The square root of a number aa, denoted by a\sqrt{a}, is a value that, when multiplied by itself, gives the original number aa. In other words, a⋅a=a\sqrt{a} \cdot \sqrt{a} = a. This property will be useful in simplifying the given expression.

Simplifying the Given Expression

Now, let's simplify the given expression 6x2⋅18x2\sqrt{6 x^2} \cdot \sqrt{18 x^2}. To do this, we can use the property of square roots mentioned earlier. We can rewrite each square root as the product of two square roots:

6x2⋅18x2=6x2⋅9x2⋅2\sqrt{6 x^2} \cdot \sqrt{18 x^2} = \sqrt{6 x^2} \cdot \sqrt{9 x^2} \cdot \sqrt{2}

Next, we can simplify each square root individually. We can rewrite 6x26 x^2 as 2⋅3x22 \cdot 3 x^2, and 9x29 x^2 as 32x23^2 x^2. This gives us:

6x2⋅18x2=2⋅3x2⋅32x2⋅2\sqrt{6 x^2} \cdot \sqrt{18 x^2} = \sqrt{2 \cdot 3 x^2} \cdot \sqrt{3^2 x^2} \cdot \sqrt{2}

Now, we can simplify each square root using the property of square roots:

2⋅3x2⋅32x2⋅2=(2⋅3)⋅(32)⋅x2⋅x2⋅2\sqrt{2 \cdot 3 x^2} \cdot \sqrt{3^2 x^2} \cdot \sqrt{2} = \sqrt{(2 \cdot 3) \cdot (3^2) \cdot x^2 \cdot x^2} \cdot \sqrt{2}

Simplifying the expression inside the square root, we get:

(2⋅3)⋅(32)⋅x2⋅x2⋅2=2⋅33⋅x4⋅2\sqrt{(2 \cdot 3) \cdot (3^2) \cdot x^2 \cdot x^2} \cdot \sqrt{2} = \sqrt{2 \cdot 3^3 \cdot x^4} \cdot \sqrt{2}

Now, we can simplify the expression further by combining the two square roots:

2⋅33⋅x4⋅2=22⋅33⋅x4\sqrt{2 \cdot 3^3 \cdot x^4} \cdot \sqrt{2} = \sqrt{2^2 \cdot 3^3 \cdot x^4}

Finally, we can simplify the expression inside the square root:

22⋅33⋅x4=2⋅33x2\sqrt{2^2 \cdot 3^3 \cdot x^4} = 2 \cdot 3 \sqrt{3 x^2}

Evaluating the Choices

Now that we have simplified the given expression, let's evaluate the choices:

A. 108x2\sqrt{108 x^2} B. 6x236 x^2 \sqrt{3} C. 63x6 \sqrt{3 x} D. 618x6 \sqrt{18 x}

To determine which choice is equivalent to the simplified expression, we can start by simplifying each choice:

A. 108x2=36⋅3x2=63x2\sqrt{108 x^2} = \sqrt{36 \cdot 3 x^2} = 6 \sqrt{3 x^2}

B. 6x23=6x236 x^2 \sqrt{3} = 6 x^2 \sqrt{3}

C. 63x=63x6 \sqrt{3 x} = 6 \sqrt{3 x}

D. 618x=69⋅2x=6⋅32x=182x6 \sqrt{18 x} = 6 \sqrt{9 \cdot 2 x} = 6 \cdot 3 \sqrt{2 x} = 18 \sqrt{2 x}

Comparing the simplified choices with the simplified expression, we can see that:

A. 63x2=2⋅33x26 \sqrt{3 x^2} = 2 \cdot 3 \sqrt{3 x^2}

B. 6x23=6x236 x^2 \sqrt{3} = 6 x^2 \sqrt{3}

C. 63x=63x6 \sqrt{3 x} = 6 \sqrt{3 x}

D. 182x=2⋅3⋅32x18 \sqrt{2 x} = 2 \cdot 3 \cdot 3 \sqrt{2 x}

From the comparison, we can see that choice A is equivalent to the simplified expression.

Conclusion

In this article, we simplified the product of two square roots, specifically 6x2⋅18x2\sqrt{6 x^2} \cdot \sqrt{18 x^2}, and determined which choice is equivalent to this product when x≥0x \geq 0. We found that choice A, 108x2\sqrt{108 x^2}, is equivalent to the simplified expression. This demonstrates the importance of simplifying radical expressions and understanding the properties of square roots.

Final Answer

Introduction

In our previous article, we simplified the product of two square roots, specifically 6x2⋅18x2\sqrt{6 x^2} \cdot \sqrt{18 x^2}, and determined which choice is equivalent to this product when x≥0x \geq 0. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in simplifying radical expressions.

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is an expression that contains a square root or other root, while a rational expression is an expression that contains a fraction. For example, 6x2\sqrt{6 x^2} is a radical expression, while 6x23x\frac{6 x^2}{3 x} is a rational expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the following steps:

  1. Simplify the expression inside the square root.
  2. Look for perfect squares that can be factored out of the expression.
  3. Use the property of square roots to simplify the expression.

Q: What is the property of square roots?

A: The property of square roots states that a⋅a=a\sqrt{a} \cdot \sqrt{a} = a. This means that when you multiply two square roots together, you can simplify the expression by combining the two square roots.

Q: How do I simplify a product of two square roots?

A: To simplify a product of two square roots, you can use the following steps:

  1. Simplify each square root individually.
  2. Use the property of square roots to simplify the expression.
  3. Combine the two simplified expressions.

Q: What is the difference between a square root and a cube root?

A: A square root is a root that is raised to the power of 1/2, while a cube root is a root that is raised to the power of 1/3. For example, 6x2\sqrt{6 x^2} is a square root, while 6x23\sqrt[3]{6 x^2} is a cube root.

Q: How do I simplify a cube root expression?

A: To simplify a cube root expression, you can use the following steps:

  1. Simplify the expression inside the cube root.
  2. Look for perfect cubes that can be factored out of the expression.
  3. Use the property of cube roots to simplify the expression.

Q: What is the property of cube roots?

A: The property of cube roots states that a3⋅a3⋅a3=a\sqrt[3]{a} \cdot \sqrt[3]{a} \cdot \sqrt[3]{a} = a. This means that when you multiply three cube roots together, you can simplify the expression by combining the three cube roots.

Q: How do I simplify a product of two cube roots?

A: To simplify a product of two cube roots, you can use the following steps:

  1. Simplify each cube root individually.
  2. Use the property of cube roots to simplify the expression.
  3. Combine the two simplified expressions.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in simplifying radical expressions. We covered topics such as the difference between radical and rational expressions, simplifying radical expressions, and simplifying products of square roots and cube roots. By following the steps outlined in this article, you should be able to simplify radical expressions with ease.

Final Answer

The final answer is that simplifying radical expressions is a crucial skill to master in mathematics, and by following the steps outlined in this article, you can simplify radical expressions with ease.