Which Choice Is Equivalent To The Quotient Shown Here When X \textgreater 0 X \ \textgreater \ 0 X \textgreater 0 ? 42 X 5 ÷ 6 X 3 \sqrt{42 X^5} \div \sqrt{6 X^3} 42 X 5 ​ ÷ 6 X 3 ​ A. X 7 X \sqrt{7} X 7 ​ B. X 2 7 X^2 \sqrt{7} X 2 7 ​ C. 7 X 2 7 X^2 7 X 2 D. 7 X 7 X 7 X

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the quotient shown in the problem statement: $\sqrt{42 x^5} \div \sqrt{6 x^3}$. We will examine each choice and determine which one is equivalent to the quotient.

Understanding the Problem

The problem requires us to simplify the quotient of two radical expressions: $\sqrt{42 x^5} \div \sqrt{6 x^3}$. To simplify this expression, we need to apply the rules of exponents and radicals.

Simplifying Radical Expressions

When simplifying radical expressions, we need to follow these steps:

1. Simplify the radicand: The radicand is the expression inside the square root. We need to simplify it by factoring out any perfect squares.
2. Apply the quotient rule: The quotient rule states that $\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}$. We can apply this rule to simplify the quotient.
3. Simplify the resulting expression: After applying the quotient rule, we need to simplify the resulting expression by combining like terms.

Simplifying the Quotient

Let's apply the steps outlined above to simplify the quotient:

$\sqrt{42 x^5} \div \sqrt{6 x^3}$

Step 1: Simplify the radicand

The radicand is $42 x^5$. We can factor out a perfect square: $42 x^5 = 6 x^3 \cdot 7 x^2$.

Step 2: Apply the quotient rule

Now that we have simplified the radicand, we can apply the quotient rule:

$\sqrt{42 x^5} \div \sqrt{6 x^3} = \sqrt{\frac{42 x^5}{6 x^3}}$

Step 3: Simplify the resulting expression

We can simplify the resulting expression by combining like terms:

$\sqrt{\frac{42 x^5}{6 x^3}} = \sqrt{\frac{7 x^2}{1}} = \sqrt{7 x^2}$

Evaluating the Choices

Now that we have simplified the quotient, we can evaluate the choices:

A. $x \sqrt{7}$

B. $x^2 \sqrt{7}$

C. $7 x^2$

D. $7 x$

Choice A: $x \sqrt{7}$

This choice is not equivalent to the quotient. We can see that the $x^2$ term is missing, which is a result of the simplification process.

Choice B: $x^2 \sqrt{7}$

This choice is equivalent to the quotient. We can see that the $x^2$ term is present, which is a result of the simplification process.

Choice C: $7 x^2$

This choice is not equivalent to the quotient. We can see that the $\sqrt{7}$ term is missing, which is a result of the simplification process.

Choice D: $7 x$

This choice is not equivalent to the quotient. We can see that the $x^2$ term is missing, which is a result of the simplification process.

Conclusion

In conclusion, the correct choice is B. $x^2 \sqrt{7}$. This choice is equivalent to the quotient, and it is the result of simplifying the radical expression using the rules of exponents and radicals.

Final Answer

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the quotient shown in the problem statement: $\sqrt{42 x^5} \div \sqrt{6 x^3}$. We will also provide a Q&A guide to help you understand the concepts and apply them to real-world problems.

Q&A Guide

Q: What is the quotient rule for radical expressions?

A: The quotient rule states that $\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}$. This rule allows us to simplify the quotient of two radical expressions by dividing the radicands.

Q: How do I simplify the radicand in a radical expression?

A: To simplify the radicand, you need to factor out any perfect squares. For example, if the radicand is $42 x^5$, you can factor out a perfect square: $42 x^5 = 6 x^3 \cdot 7 x^2$.

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

A: A perfect square is a number that can be expressed as the square of an integer, such as $4 = 2^2$ or $9 = 3^2$. A perfect cube is a number that can be expressed as the cube of an integer, such as $8 = 2^3$ or $27 = 3^3$.

Q: How do I apply the quotient rule to simplify a radical expression?

A: To apply the quotient rule, you need to divide the radicands and simplify the resulting expression. For example, if you have the expression $\sqrt{42 x^5} \div \sqrt{6 x^3}$, you can apply the quotient rule as follows:

$\sqrt{42 x^5} \div \sqrt{6 x^3} = \sqrt{\frac{42 x^5}{6 x^3}}$

Q: What is the resulting expression after applying the quotient rule?

A: The resulting expression is $\sqrt{7 x^2}$. This expression is the simplified form of the original radical expression.

Q: How do I simplify the resulting expression further?

A: To simplify the resulting expression further, you can combine like terms. For example, if you have the expression $\sqrt{7 x^2}$, you can simplify it as follows:

$\sqrt{7 x^2} = \sqrt{7} \cdot \sqrt{x^2} = \sqrt{7} \cdot x$

Q: What is the final simplified form of the radical expression?

A: The final simplified form of the radical expression is $\sqrt{7} \cdot x$. This expression is the result of simplifying the original radical expression using the rules of exponents and radicals.

Common Mistakes to Avoid

• Not simplifying the radicand: Failing to simplify the radicand can lead to incorrect results.
• Not applying the quotient rule: Failing to apply the quotient rule can lead to incorrect results.
• Not simplifying the resulting expression: Failing to simplify the resulting expression can lead to incorrect results.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By understanding the rules of exponents and radicals, you can simplify radical expressions and apply them to real-world problems. Remember to simplify the radicand, apply the quotient rule, and simplify the resulting expression to get the final answer.

Final Tips

• Practice, practice, practice: The more you practice simplifying radical expressions, the more comfortable you will become with the rules and procedures.
• Use online resources: There are many online resources available that can help you learn and practice simplifying radical expressions.
• Seek help when needed: Don't be afraid to ask for help if you are struggling with a particular problem or concept.