If X ≠ 0 X \neq 0 X  = 0 , What Is The Sum Of 4 X 10 3 + 5 X 3 8 X 3 4 \sqrt[3]{x^{10}} + 5 X^3 \sqrt[3]{8 X} 4 3 X 10 ​ + 5 X 3 3 8 X ​ In Simplest Form? Enter The Correct Answer In The Box.

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Understanding the Problem

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When dealing with radical expressions, it's essential to simplify them to their most basic form. In this case, we're given the expression $4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x}$ and asked to find its sum in simplest form. To tackle this problem, we'll need to apply the properties of radicals and exponents.

Breaking Down the Expression

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Let's start by breaking down the given expression into its individual components:

• $4 \sqrt[3]{x^{10}}$
• $5 x^3 \sqrt[3]{8 x}$

We can simplify each component separately and then combine them to find the final sum.

Simplifying the First Component

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The first component is $4 \sqrt[3]{x^{10}}$. To simplify this expression, we can use the property of radicals that states $\sqrt[n]{a^n} = a$. In this case, we have $\sqrt[3]{x^{10}}$, which can be rewritten as $\sqrt[3]{x^3 \cdot x^7}$. Using the property mentioned earlier, we can simplify this expression to $x \sqrt[3]{x^7}$.

Now, we can multiply this simplified expression by 4 to get the final result for the first component: $4x \sqrt[3]{x^7}$.

Simplifying the Second Component

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The second component is $5 x^3 \sqrt[3]{8 x}$. To simplify this expression, we can start by simplifying the radical term $\sqrt[3]{8 x}$. We can rewrite 8 as $2^3$, so the expression becomes $\sqrt[3]{2^3 x}$. Using the property of radicals mentioned earlier, we can simplify this expression to $2 \sqrt[3]{x}$.

Now, we can multiply this simplified expression by $5 x^3$ to get the final result for the second component: $10 x^3 \sqrt[3]{x}$.

Combining the Components

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Now that we've simplified both components, we can combine them to find the final sum:

$4x \sqrt[3]{x^7} + 10 x^3 \sqrt[3]{x}$

To combine these expressions, we can start by factoring out the common term $\sqrt[3]{x}$ from both components. This gives us:

$\sqrt[3]{x} (4x \sqrt[3]{x^6} + 10 x^3)$

Now, we can simplify the expression inside the parentheses by combining like terms:

$4x^4 + 10 x^3$

Factoring out the common term $2x^3$, we get:

$2x^3 (2x + 5)$

The Final Answer

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Therefore, the sum of $4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x}$ in simplest form is:

$2x^3 (2x + 5)$

This is the final answer to the problem.

Conclusion

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Simplifying radical expressions can be a challenging task, but by applying the properties of radicals and exponents, we can break down complex expressions into their most basic form. In this case, we started with the expression $4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x}$ and simplified it to its final form: $2x^3 (2x + 5)$. By following these steps, you can simplify any radical expression and find its sum in simplest form.

Key Takeaways

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• To simplify a radical expression, start by breaking it down into its individual components.
• Use the properties of radicals and exponents to simplify each component separately.
• Combine the simplified components to find the final sum.
• Factor out common terms to simplify the expression further.

By following these key takeaways, you can simplify any radical expression and find its sum in simplest form.

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

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A: A radical is a mathematical operation that involves finding the root of a number, while an exponent is a mathematical operation that involves raising a number to a power. For example, $\sqrt{x}$ is a radical, while $x^2$ is an exponent.

Q: How do I simplify a radical expression?

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A: To simplify a radical expression, start by breaking it down into its individual components. Use the properties of radicals and exponents to simplify each component separately. Combine the simplified components to find the final sum. Factor out common terms to simplify the expression further.

Q: What is the property of radicals that states $\sqrt[n]{a^n} = a$?

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A: This property states that when a number is raised to a power and then taken to the nth root, the result is equal to the original number. For example, $\sqrt[3]{x^3} = x$.

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

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A: To simplify a radical expression with multiple terms, start by simplifying each term separately. Use the properties of radicals and exponents to simplify each term. Combine the simplified terms to find the final sum. Factor out common terms to simplify the expression further.

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

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A: A cube root is a mathematical operation that involves finding the third root of a number, while a square root is a mathematical operation that involves finding the second root of a number. For example, $\sqrt[3]{x}$ is a cube root, while $\sqrt{x}$ is a square root.

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

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A: To simplify a radical expression with a coefficient, start by simplifying the radical term. Use the properties of radicals and exponents to simplify the radical term. Then, multiply the simplified radical term by the coefficient to find the final result.

Q: What is the final answer to the problem $4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x}$?

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A: The final answer to the problem $4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x}$ is $2x^3 (2x + 5)$.

Q: How do I know if a radical expression is in simplest form?

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A: A radical expression is in simplest form when it cannot be simplified further using the properties of radicals and exponents. To check if a radical expression is in simplest form, start by simplifying each term separately. If the expression cannot be simplified further, then it is in simplest form.

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

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

• Not simplifying each term separately
• Not using the properties of radicals and exponents correctly
• Not factoring out common terms
• Not checking if the expression is in simplest form

By avoiding these common mistakes, you can ensure that your radical expressions are simplified correctly.

Conclusion

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Simplifying radical expressions can be a challenging task, but by following the steps outlined in this article, you can simplify any radical expression and find its sum in simplest form. Remember to break down the expression into its individual components, use the properties of radicals and exponents to simplify each component, and combine the simplified components to find the final sum. By following these steps, you can simplify any radical expression and find its sum in simplest form.