Which Choice Is Equivalent To The Expression Below When $x \geq 0$?$\sqrt{50 X^3} - \sqrt{25 X^3} + 5 \sqrt{x^3} - \sqrt{2 X^3}$A. $5 \sqrt{2 X}$ B. $4 \sqrt{x}$ C. $4 X \sqrt{2 X}$ D. $28

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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 how to simplify a given radical expression, 50x3−25x3+5x3−2x3\sqrt{50 x^3} - \sqrt{25 x^3} + 5 \sqrt{x^3} - \sqrt{2 x^3}, when x≥0x \geq 0. We will break down the expression into smaller parts, simplify each part, and then combine them to arrive at the final answer.

Step 1: Simplify Each Radical Expression

To simplify the given expression, we need to start by simplifying each radical expression individually.

Simplifying 50x3\sqrt{50 x^3}

We can simplify 50x3\sqrt{50 x^3} by factoring out the perfect square from under the square root sign.

50x3=25â‹…2â‹…x3=25x3â‹…2=5x2x\sqrt{50 x^3} = \sqrt{25 \cdot 2 \cdot x^3} = \sqrt{25 x^3} \cdot \sqrt{2} = 5 x \sqrt{2 x}

Simplifying 25x3\sqrt{25 x^3}

We can simplify 25x3\sqrt{25 x^3} by factoring out the perfect square from under the square root sign.

25x3=25â‹…x3=5xx\sqrt{25 x^3} = \sqrt{25 \cdot x^3} = 5 x \sqrt{x}

Simplifying 5x35 \sqrt{x^3}

We can simplify 5x35 \sqrt{x^3} by factoring out the perfect square from under the square root sign.

5x3=5xx5 \sqrt{x^3} = 5 x \sqrt{x}

Simplifying 2x3\sqrt{2 x^3}

We can simplify 2x3\sqrt{2 x^3} by factoring out the perfect square from under the square root sign.

2x3=2â‹…x3=2â‹…xx\sqrt{2 x^3} = \sqrt{2} \cdot \sqrt{x^3} = \sqrt{2} \cdot x \sqrt{x}

Step 2: Combine the Simplified Expressions

Now that we have simplified each radical expression, we can combine them to arrive at the final answer.

50x3−25x3+5x3−2x3=5x2x−5xx+5xx−2⋅xx\sqrt{50 x^3} - \sqrt{25 x^3} + 5 \sqrt{x^3} - \sqrt{2 x^3} = 5 x \sqrt{2 x} - 5 x \sqrt{x} + 5 x \sqrt{x} - \sqrt{2} \cdot x \sqrt{x}

We can simplify this expression further by combining like terms.

5x2x−5xx+5xx−2⋅xx=5x2x−2⋅xx5 x \sqrt{2 x} - 5 x \sqrt{x} + 5 x \sqrt{x} - \sqrt{2} \cdot x \sqrt{x} = 5 x \sqrt{2 x} - \sqrt{2} \cdot x \sqrt{x}

We can factor out the common term, xxx \sqrt{x}, from both terms.

5x2x−2⋅xx=xx(52−2)5 x \sqrt{2 x} - \sqrt{2} \cdot x \sqrt{x} = x \sqrt{x} (5 \sqrt{2} - \sqrt{2})

We can simplify this expression further by combining like terms.

xx(52−2)=xx⋅42x \sqrt{x} (5 \sqrt{2} - \sqrt{2}) = x \sqrt{x} \cdot 4 \sqrt{2}

We can simplify this expression further by combining like terms.

xxâ‹…42=4x2xx \sqrt{x} \cdot 4 \sqrt{2} = 4 x \sqrt{2 x}

Conclusion

In this article, we simplified the given radical expression, 50x3−25x3+5x3−2x3\sqrt{50 x^3} - \sqrt{25 x^3} + 5 \sqrt{x^3} - \sqrt{2 x^3}, when x≥0x \geq 0. We broke down the expression into smaller parts, simplified each part, and then combined them to arrive at the final answer, 4x2x4 x \sqrt{2 x}.

Answer

The correct answer is:

  • C. 4x2x4 x \sqrt{2 x}

Discussion

This problem requires a good understanding of radical expressions and simplifying them. The key to solving this problem is to simplify each radical expression individually and then combine them to arrive at the final answer. The correct answer, 4x2x4 x \sqrt{2 x}, is the result of simplifying the given expression.

Additional Tips

  • When simplifying radical expressions, it is essential to factor out the perfect square from under the square root sign.
  • When combining like terms, it is essential to combine the coefficients and the variables separately.
  • When simplifying radical expressions, it is essential to simplify each expression individually and then combine them to arrive at the final answer.

References

Conclusion

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or a higher root of a number or expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to factor out the perfect square from under the square root sign, combine like terms, and simplify each expression individually before combining them to arrive at the final answer.

Q: What is a perfect square?

A: A perfect square is a number or expression that can be expressed as the square of an integer or a variable.

Q: How do I factor out a perfect square from under the square root sign?

A: To factor out a perfect square from under the square root sign, you need to identify the perfect square and multiply it by the square root of the remaining expression.

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

A: A radical expression is an expression that contains a square root or a higher root of a number or expression, while an exponential expression is an expression that contains a power of a number or expression.

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

A: No, you cannot simplify a radical expression with a negative exponent. The exponent must be a positive integer.

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

A: To simplify a radical expression with a variable in the radicand, you need to factor out the perfect square from under the square root sign, combine like terms, and simplify each expression individually before combining them to arrive at the final answer.

Q: Can I simplify a radical expression with a fraction in the radicand?

A: Yes, you can simplify a radical expression with a fraction in the radicand by factoring out the perfect square from under the square root sign, combining like terms, and simplifying each expression individually before combining them to arrive at the final answer.

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

A: To simplify a radical expression with a decimal in the radicand, you need to round the decimal to the nearest integer and then simplify the radical expression.

Q: Can I simplify a radical expression with a negative number in the radicand?

A: No, you cannot simplify a radical expression with a negative number in the radicand. The radicand must be a non-negative number.

Q: How do I simplify a radical expression with a variable in the denominator?

A: To simplify a radical expression with a variable in the denominator, you need to factor out the perfect square from under the square root sign, combine like terms, and simplify each expression individually before combining them to arrive at the final answer.

Q: Can I simplify a radical expression with a fraction in the denominator?

A: Yes, you can simplify a radical expression with a fraction in the denominator by factoring out the perfect square from under the square root sign, combining like terms, and simplifying each expression individually before combining them to arrive at the final answer.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By following the steps outlined in this article, you can simplify even the most complex radical expressions. Remember to factor out the perfect square from under the square root sign, combine like terms, and simplify each expression individually before combining them to arrive at the final answer.

Additional Tips

  • When simplifying radical expressions, it is essential to factor out the perfect square from under the square root sign.
  • When combining like terms, it is essential to combine the coefficients and the variables separately.
  • When simplifying radical expressions, it is essential to simplify each expression individually and then combine them to arrive at the final answer.

References

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By following the steps outlined in this article, you can simplify even the most complex radical expressions. Remember to factor out the perfect square from under the square root sign, combine like terms, and simplify each expression individually before combining them to arrive at the final answer.