Which Choice Is Equivalent To The Quotient Shown Here When X \textgreater 0 X \ \textgreater \ 0 X \textgreater 0 ? 18 X 3 ÷ 9 X \sqrt{18 X^3} \div \sqrt{9 X} 18 X 3 ​ ÷ 9 X ​ A. 2 X 2 X 2 X B. X 2 2 X^2 \sqrt{2} X 2 2 ​ C. 2 X \sqrt{2 X} 2 X ​ D. X 2 X \sqrt{2} X 2 ​

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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: 18x3÷9x\sqrt{18 x^3} \div \sqrt{9 x}. We will examine each choice and determine which one is equivalent to the quotient.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or other root. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Simplifying Radical Expressions

To simplify a radical expression, we need to follow a series of steps:

  1. Factor the radicand: The radicand is the number inside the square root. We need to factor the radicand into its prime factors.
  2. Identify the square root of perfect squares: 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 squared.
  3. Simplify the expression: Once we have identified the square root of perfect squares, we can simplify the expression by removing the square root.

Simplifying the Quotient

Now, let's apply these steps to the quotient shown in the problem statement: 18x3÷9x\sqrt{18 x^3} \div \sqrt{9 x}.

Step 1: Factor the Radicand

The radicand is 18x318 x^3. We can factor this expression as follows:

18x3=2⋅32⋅x2⋅x18 x^3 = 2 \cdot 3^2 \cdot x^2 \cdot x

Step 2: Identify the Square Root of Perfect Squares

The expression 323^2 is a perfect square because it can be expressed as the square of an integer (3). Therefore, we can simplify the expression as follows:

18x3=2⋅32⋅x2⋅x=2⋅3⋅x⋅x2\sqrt{18 x^3} = \sqrt{2 \cdot 3^2 \cdot x^2 \cdot x} = \sqrt{2} \cdot 3 \cdot x \cdot \sqrt{x^2}

Step 3: Simplify the Expression

The expression x2\sqrt{x^2} is equal to xx because the square root of a perfect square is equal to the number inside the square root. Therefore, we can simplify the expression as follows:

18x3=2⋅3⋅x⋅x=2⋅3⋅x2\sqrt{18 x^3} = \sqrt{2} \cdot 3 \cdot x \cdot x = \sqrt{2} \cdot 3 \cdot x^2

Step 4: Simplify the Quotient

Now, let's simplify the quotient by dividing the two expressions:

18x39x=2⋅3⋅x29⋅x\frac{\sqrt{18 x^3}}{\sqrt{9 x}} = \frac{\sqrt{2} \cdot 3 \cdot x^2}{\sqrt{9} \cdot \sqrt{x}}

Step 5: Simplify the Expression

The expression 9\sqrt{9} is equal to 3 because the square root of a perfect square is equal to the number inside the square root. Therefore, we can simplify the expression as follows:

18x39x=2⋅3⋅x23⋅x\frac{\sqrt{18 x^3}}{\sqrt{9 x}} = \frac{\sqrt{2} \cdot 3 \cdot x^2}{3 \cdot \sqrt{x}}

Step 6: Simplify the Expression

We can simplify the expression further by canceling out the common factors:

18x39x=2⋅x2x\frac{\sqrt{18 x^3}}{\sqrt{9 x}} = \frac{\sqrt{2} \cdot x^2}{\sqrt{x}}

Step 7: Simplify the Expression

The expression x\sqrt{x} is equal to xx because the square root of a perfect square is equal to the number inside the square root. Therefore, we can simplify the expression as follows:

18x39x=2⋅x⋅x=2⋅x2\frac{\sqrt{18 x^3}}{\sqrt{9 x}} = \sqrt{2} \cdot x \cdot x = \sqrt{2} \cdot x^2

Conclusion

In conclusion, the quotient 18x3÷9x\sqrt{18 x^3} \div \sqrt{9 x} is equivalent to 2⋅x2\sqrt{2} \cdot x^2. This is the correct answer.

Answer Key

The correct answer is:

  • B. x22x^2 \sqrt{2}

Discussion

This problem requires a deep understanding of radical expressions and simplification techniques. The student must be able to factor the radicand, identify the square root of perfect squares, and simplify the expression. The student must also be able to cancel out common factors and simplify the expression further.

Tips and Tricks

Here are some tips and tricks to help you solve this problem:

  • Make sure to factor the radicand completely.
  • Identify the square root of perfect squares and simplify the expression.
  • Cancel out common factors and simplify the expression further.
  • Use the properties of square roots to simplify the expression.

Practice Problems

Here are some practice problems to help you reinforce your understanding of radical expressions and simplification techniques:

  • Simplify the expression: 20x4÷5x2\sqrt{20 x^4} \div \sqrt{5 x^2}
  • Simplify the expression: 24x3÷6x\sqrt{24 x^3} \div \sqrt{6 x}
  • Simplify the expression: 36x4÷9x2\sqrt{36 x^4} \div \sqrt{9 x^2}
    Radical Expressions Q&A ==========================

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or other root. The square root of a number is a value that, when multiplied by itself, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow a series of steps:

  1. Factor the radicand: The radicand is the number inside the square root. You need to factor the radicand into its prime factors.
  2. Identify the square root of perfect squares: 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 squared.
  3. Simplify the expression: Once you have identified the square root of perfect squares, you can simplify the expression by removing the square root.

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. For example, 16 is a perfect square because it can be expressed as 4 squared. A perfect cube is a number that can be expressed as the cube of an integer. For example, 27 is a perfect cube because it can be expressed as 3 cubed.

Q: How do I simplify a radical expression with a perfect cube?

A: To simplify a radical expression with a perfect cube, you need to follow the same steps as before:

  1. Factor the radicand: The radicand is the number inside the square root. You need to factor the radicand into its prime factors.
  2. Identify the square root of perfect squares: 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 squared.
  3. Simplify the expression: Once you have identified the square root of perfect squares, you can simplify the expression by removing the square root.

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

A: A rational expression is a mathematical expression that contains a fraction. A radical expression is a mathematical expression that contains a square root or other root.

Q: How do I simplify a rational expression with a radical?

A: To simplify a rational expression with a radical, you need to follow the same steps as before:

  1. Simplify the radical expression: Simplify the radical expression by following the steps outlined above.
  2. Simplify the fraction: Simplify the fraction by canceling out common factors.

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

A: A radical expression is a mathematical expression that contains a square root or other root. An exponential expression is a mathematical expression that contains a base raised to a power.

Q: How do I simplify a radical expression with an exponential expression?

A: To simplify a radical expression with an exponential expression, you need to follow the same steps as before:

  1. Simplify the radical expression: Simplify the radical expression by following the steps outlined above.
  2. Simplify the exponential expression: Simplify the exponential expression by following the rules of exponents.

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

A: A radical expression is a mathematical expression that contains a square root or other root. A trigonometric expression is a mathematical expression that contains trigonometric functions such as sine, cosine, and tangent.

Q: How do I simplify a radical expression with a trigonometric expression?

A: To simplify a radical expression with a trigonometric expression, you need to follow the same steps as before:

  1. Simplify the radical expression: Simplify the radical expression by following the steps outlined above.
  2. Simplify the trigonometric expression: Simplify the trigonometric expression by following the rules of trigonometry.

Conclusion

In conclusion, radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. By following the steps outlined above, you can simplify radical expressions and solve a wide range of mathematical problems. Remember to factor the radicand, identify the square root of perfect squares, and simplify the expression to get the final answer.