Select The Correct Answer.Which Expression Is Equivalent To $8 X^2 \sqrt[3]{375 X} + 2 \sqrt[3]{3 X^7}$, If X ≠ 0 X \neq 0 X  = 0 ?A. 10 X 4 125 X 3 10 X^4 \sqrt[3]{125 X} 10 X 4 3 125 X ​ B. 42 X 3 42 X^3 42 X 3 C. 42 X 2 3 X 3 42 X^2 \sqrt[3]{3 X} 42 X 2 3 3 X ​ D. $10 X^2

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Introduction

Radical expressions can be complex and challenging to simplify, but with the right techniques and strategies, they can be broken down into more manageable components. In this article, we will explore how to simplify radical expressions, focusing on the given expression $8 x^2 \sqrt[3]{375 x} + 2 \sqrt[3]{3 x^7}$ and determine which of the provided options is equivalent to it.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a power of a number. In this case, we are dealing with a cube root, denoted by the symbol x3\sqrt[3]{x}. The cube root of a number is a value that, when multiplied by itself twice, gives the original number.

Breaking Down the Given Expression

The given expression is $8 x^2 \sqrt[3]{375 x} + 2 \sqrt[3]{3 x^7}.$ To simplify this expression, we need to break it down into its individual components and then manipulate them to obtain a simpler form.

Step 1: Factorize the Numbers Inside the Cube Roots

The first step is to factorize the numbers inside the cube roots. We can factorize 375 as 3⋅533 \cdot 5^3 and 3x73 x^7 as 3x6⋅x3 x^6 \cdot x.

import math

x = 1  # assume x is a positive number



num1 = 375
num2 = 3 * x**7



factor1 = 3 * (5**3)



factor2 = 3 * x**6 * x

Step 2: Simplify the Cube Roots

Now that we have factorized the numbers inside the cube roots, we can simplify them. We can rewrite the cube roots as follows:

375x3=3⋅53⋅x3=5x33\sqrt[3]{375 x} = \sqrt[3]{3 \cdot 5^3 \cdot x} = 5 x \sqrt[3]{3}

3x73=3x6⋅x3=x23x3\sqrt[3]{3 x^7} = \sqrt[3]{3 x^6 \cdot x} = x^2 \sqrt[3]{3 x} 

# Simplify the cube roots
cube_root1 = 5 * x * (3 ** (1/3))
cube_root2 = x**2 * (3 ** (1/3)) * x

Step 3: Substitute the Simplified Cube Roots Back into the Original Expression

Now that we have simplified the cube roots, we can substitute them back into the original expression:

8x2375x3+23x73=8x2(5x33)+2(x23x3)8 x^2 \sqrt[3]{375 x} + 2 \sqrt[3]{3 x^7} = 8 x^2 (5 x \sqrt[3]{3}) + 2 (x^2 \sqrt[3]{3 x}) 

# Substitute the simplified cube roots back into the original expression
expression = 8 * x**2 * cube_root1 + 2 * cube_root2

Step 4: Simplify the Expression

Now that we have substituted the simplified cube roots back into the original expression, we can simplify it further:

8x2(5x33)+2(x23x3)=40x333+2x23x38 x^2 (5 x \sqrt[3]{3}) + 2 (x^2 \sqrt[3]{3 x}) = 40 x^3 \sqrt[3]{3} + 2 x^2 \sqrt[3]{3 x} 

# Simplify the expression
simplified_expression = 40 * x**3 * (3 ** (1/3)) + 2 * x**2 * (3 ** (1/3)) * x

Step 5: Factor Out the Common Terms

Now that we have simplified the expression, we can factor out the common terms:

40x333+2x23x3=2x2(20x33+3x3)40 x^3 \sqrt[3]{3} + 2 x^2 \sqrt[3]{3 x} = 2 x^2 (20 x \sqrt[3]{3} + \sqrt[3]{3 x}) 

# Factor out the common terms
factored_expression = 2 * x**2 * (20 * x * (3 ** (1/3)) + (3 ** (1/3)) * x)

Step 6: Simplify the Factored Expression

Now that we have factored out the common terms, we can simplify the factored expression:

2x2(20x33+3x3)=2x233(20x+x2/3)2 x^2 (20 x \sqrt[3]{3} + \sqrt[3]{3 x}) = 2 x^2 \sqrt[3]{3} (20 x + x^{2/3}) 

# Simplify the factored expression
simplified_factored_expression = 2 * x**2 * (3 ** (1/3)) * (20 * x + x**(2/3))

Step 7: Rewrite the Expression in a Simpler Form

Now that we have simplified the factored expression, we can rewrite it in a simpler form:

2x233(20x+x2/3)=2x233(20x+x2/3)2 x^2 \sqrt[3]{3} (20 x + x^{2/3}) = 2 x^2 \sqrt[3]{3} (20 x + x^{2/3}) 

# Rewrite the expression in a simpler form
simplified_expression = 2 * x**2 * (3 ** (1/3)) * (20 * x + x**(2/3))

Step 8: Compare the Simplified Expression with the Options

Now that we have simplified the expression, we can compare it with the options:

A. 10x4125x310 x^4 \sqrt[3]{125 x}

B. 42x342 x^3

C. 42x23x342 x^2 \sqrt[3]{3 x}

D. 10x23x310 x^2 \sqrt[3]{3 x}

# Compare the simplified expression with the options
options = [
    "10 x^4 sqrt[3](125 x)",
    "42 x^3",
    "42 x^2 sqrt[3](3 x)",
    "10 x^2 sqrt[3](3 x)"
]


print("Simplified Expression:", simplified_expression)



for i, option in enumerate(options):
print(f"Option i+1} {option")

Conclusion

In conclusion, we have simplified the given expression $8 x^2 \sqrt[3]{375 x} + 2 \sqrt[3]{3 x^7}$ and compared it with the options. The simplified expression is 2x233(20x+x2/3)2 x^2 \sqrt[3]{3} (20 x + x^{2/3}), which is equivalent to option C. 42x23x342 x^2 \sqrt[3]{3 x}.

Final Answer

Introduction

Radical expressions can be complex and challenging to simplify, but with the right techniques and strategies, they can be broken down into more manageable components. In this article, we will explore some common questions and answers related to simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a power of a number. In this case, we are dealing with a cube root, denoted by the symbol x3\sqrt[3]{x}. The cube root of a number is a value that, when multiplied by itself twice, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break it down into its individual components and then manipulate them to obtain a simpler form. Here are the steps to follow:

  1. Factorize the numbers inside the cube roots.
  2. Simplify the cube roots.
  3. Substitute the simplified cube roots back into the original expression.
  4. Simplify the expression further.
  5. Factor out the common terms.
  6. Simplify the factored expression.
  7. Rewrite the expression in a simpler form.

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

A: A cube root is a root that is raised to the power of 3, while a square root is a root that is raised to the power of 2. In other words, a cube root is a value that, when multiplied by itself twice, gives the original number, while a square root is a value that, when multiplied by itself once, gives the original number.

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

A: To simplify a radical expression with multiple terms, you need to follow the same steps as before, but you also need to combine like terms. Here are the steps to follow:

  1. Factorize the numbers inside the cube roots.
  2. Simplify the cube roots.
  3. Substitute the simplified cube roots back into the original expression.
  4. Simplify the expression further.
  5. Factor out the common terms.
  6. Simplify the factored expression.
  7. Rewrite the expression in a simpler form.
  8. Combine like terms.

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

A: A radical expression is a mathematical expression that contains a root or a power of a number, while an exponential expression is a mathematical expression that contains a base and an exponent. In other words, a radical expression is a value that, when multiplied by itself a certain number of times, gives the original number, while an exponential expression is a value that, when multiplied by itself a certain number of times, gives a new value.

Q: How do I simplify a radical expression with a negative exponent?

A: To simplify a radical expression with a negative exponent, you need to follow the same steps as before, but you also need to use the rule that a−n=1ana^{-n} = \frac{1}{a^n}. Here are the steps to follow:

  1. Factorize the numbers inside the cube roots.
  2. Simplify the cube roots.
  3. Substitute the simplified cube roots back into the original expression.
  4. Simplify the expression further.
  5. Factor out the common terms.
  6. Simplify the factored expression.
  7. Rewrite the expression in a simpler form.
  8. Use the rule that a−n=1ana^{-n} = \frac{1}{a^n}.

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

A: A radical expression is a mathematical expression that contains a root or a power of a number, while a rational expression is a mathematical expression that contains a fraction. In other words, a radical expression is a value that, when multiplied by itself a certain number of times, gives the original number, while a rational expression is a value that is a fraction of two numbers.

Conclusion

In conclusion, simplifying radical expressions can be a complex and challenging task, but with the right techniques and strategies, it can be broken down into more manageable components. By following the steps outlined in this article, you can simplify radical expressions and make them easier to work with.

Final Answer

The final answer is C. 42x23x342 x^2 \sqrt[3]{3 x}.