Select The Correct Answer.Which Expression Is Equivalent To 2 X 44 X − 2 11 X 3 2x\sqrt{44x} - 2\sqrt{11x^3} 2 X 44 X ​ − 2 11 X 3 ​ , If X \textgreater 0 X \ \textgreater \ 0 X \textgreater 0 ?A. 2 X 11 X 2x\sqrt{11x} 2 X 11 X ​ B. 6 X 22 X 6x\sqrt{22x} 6 X 22 X ​ C. 2 X 2x 2 X D. 8 X 2 11 8x^2\sqrt{11} 8 X 2 11 ​

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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 given expression 2x44x−211x32x\sqrt{44x} - 2\sqrt{11x^3}, and determine which of the provided options is equivalent to it.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher-order 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.

Radical expressions can be simplified by factoring out perfect squares from the radicand (the number inside the radical sign). This process involves identifying the largest perfect square that divides the radicand and then taking the square root of that perfect square.

Simplifying the Given Expression

The given expression is 2x44x−211x32x\sqrt{44x} - 2\sqrt{11x^3}. To simplify this expression, we need to factor out perfect squares from the radicands.

Step 1: Factor out Perfect Squares

The radicand 44x44x can be factored as 4⋅11⋅x4 \cdot 11 \cdot x. Since 44 is a perfect square, we can rewrite the expression as 2x4⋅11⋅x2x\sqrt{4 \cdot 11 \cdot x}.

Step 2: Simplify the Square Root

The square root of 44 is 22, so we can simplify the expression as 2x⋅211x2x \cdot 2 \sqrt{11x}, which equals 4x11x4x\sqrt{11x}.

Step 3: Simplify the Second Term

The radicand 11x311x^3 can be factored as 11⋅x2⋅x11 \cdot x^2 \cdot x. Since x2x^2 is a perfect square, we can rewrite the expression as 211⋅x2⋅x2\sqrt{11 \cdot x^2 \cdot x}.

Step 4: Simplify the Square Root

The square root of x2x^2 is xx, so we can simplify the expression as 2x11x2x\sqrt{11x}.

Combining the Simplified Terms

Now that we have simplified both terms, we can combine them to get the final expression. However, we need to be careful when combining like terms.

Step 1: Combine Like Terms

The simplified expression is 4x11x−2x11x4x\sqrt{11x} - 2x\sqrt{11x}. Since both terms have the same radicand, we can combine them by subtracting the coefficients.

Step 2: Simplify the Result

The resulting expression is 2x11x2x\sqrt{11x}.

Conclusion

In conclusion, the expression 2x44x−211x32x\sqrt{44x} - 2\sqrt{11x^3} can be simplified to 2x11x2x\sqrt{11x}, which is equivalent to option A.

Final Answer

The correct answer is:

  • A. 2x11x2x\sqrt{11x}

Why is this the Correct Answer?

This is the correct answer because we simplified the given expression by factoring out perfect squares and combining like terms. The resulting expression matches option A, which is 2x11x2x\sqrt{11x}.

What is the Significance of this Problem?

This problem is significant because it requires the application of mathematical concepts, such as factoring and simplifying radical expressions. It also requires critical thinking and problem-solving skills to arrive at the correct answer.

What are the Key Takeaways from this Problem?

The key takeaways from this problem are:

  • The importance of factoring out perfect squares from radical expressions.
  • The process of simplifying radical expressions by combining like terms.
  • The significance of critical thinking and problem-solving skills in mathematics.

How can this Problem be Applied in Real-Life Situations?

This problem can be applied in real-life situations, such as:

  • Simplifying complex mathematical expressions in physics and engineering.
  • Solving problems in finance and economics that involve radical expressions.
  • Developing critical thinking and problem-solving skills in various fields.

What are the Limitations of this Problem?

The limitations of this problem are:

  • The problem assumes a basic understanding of mathematical concepts, such as factoring and simplifying radical expressions.
  • The problem may not be applicable in real-life situations that involve more complex mathematical expressions.
  • The problem may not be suitable for students who struggle with mathematical concepts or critical thinking skills.

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Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher-order 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 factor out perfect squares from the radicand (the number inside the radical sign). This process involves identifying the largest perfect square that divides the radicand and then taking the square root of that perfect square.

Q: What is a perfect square?

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

Q: How do I identify perfect squares in a radical expression?

A: To identify perfect squares in a radical expression, you need to look for numbers that can be expressed as the product of an integer and itself. You can also use the fact that a perfect square always has an even exponent.

Q: Can I simplify a radical expression by combining like terms?

A: Yes, you can simplify a radical expression by combining like terms. However, you need to be careful when combining like terms, as the coefficients of the terms may be different.

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 a higher-order root, while an exponential expression is a mathematical expression that contains a power or an exponent.

Q: Can I simplify a radical expression by using the properties of exponents?

A: Yes, you can simplify a radical expression by using the properties of exponents. For example, you can use the property that a^(m+n) = a^m x a^n to simplify a radical expression.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

  • Not factoring out perfect squares from the radicand.
  • Not combining like terms correctly.
  • Not using the properties of exponents correctly.
  • Not checking the final answer for errors.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through examples and exercises in a math textbook or online resource. You can also try simplifying radical expressions on your own, using a calculator or computer program to check your answers.

Q: What are some real-life applications of simplifying radical expressions?

A: Simplifying radical expressions has many real-life applications, including:

  • Simplifying complex mathematical expressions in physics and engineering.
  • Solving problems in finance and economics that involve radical expressions.
  • Developing critical thinking and problem-solving skills in various fields.

Q: Can I use technology to simplify radical expressions?

A: Yes, you can use technology to simplify radical expressions. Many calculators and computer programs, such as Mathematica or Maple, have built-in functions for simplifying radical expressions.

Q: What are some common types of radical expressions?

A: Some common types of radical expressions include:

  • Square roots (e.g. √x)
  • Cube roots (e.g. ∛x)
  • Higher-order roots (e.g. 4√x)
  • Radical expressions with multiple terms (e.g. √x + √y)

Q: How do I simplify a radical expression with multiple terms?

A: To simplify a radical expression with multiple terms, you need to factor out perfect squares from each term and then combine like terms.

Q: Can I simplify a radical expression with a negative exponent?

A: Yes, you can simplify a radical expression with a negative exponent by using the property that a^(-m) = 1/a^m.

Q: What are some common mistakes to avoid when simplifying radical expressions with negative exponents?

A: Some common mistakes to avoid when simplifying radical expressions with negative exponents include:

  • Not using the property that a^(-m) = 1/a^m correctly.
  • Not factoring out perfect squares from the radicand.
  • Not combining like terms correctly.

Q: How can I check my answers when simplifying radical expressions?

A: You can check your answers when simplifying radical expressions by:

  • Using a calculator or computer program to simplify the expression.
  • Checking the final answer for errors.
  • Working through examples and exercises in a math textbook or online resource.