If $x \neq 0$, What Is The Sum Of $4 \sqrt[3]{x^{10}}+5 X^3 \sqrt[3]{8 X}$ In Simplest Form?

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Understanding the Problem

The given problem involves simplifying an algebraic expression that contains cube roots and powers of a variable x. We are required to find the sum of the given expression in its simplest form.

Breaking Down the Expression

To simplify the given expression, we need to break it down into its individual components and then simplify each component separately.

Simplifying the First Component

The first component of the expression is $4 \sqrt[3]{x^{10}}$. We can simplify this component by using the property of cube roots that states $\sqrt[3]{a^3} = a$. Since $x^{10} = (x3)3 \cdot x$, we can rewrite the first component as $4 \sqrt[3]{(x3)3 \cdot x}$.

Simplifying the Second Component

The second component of the expression is $5 x^3 \sqrt[3]{8 x}$. We can simplify this component by using the property of cube roots that states $\sqrt[3]{a^3} = a$. Since $8 = 2^3$, we can rewrite the second component as $5 x^3 \sqrt[3]{2^3 \cdot x}$.

Combining the Components

Now that we have simplified the individual components, we can combine them to get the simplified form of the given expression.

Simplifying the Expression

We can simplify the expression by combining the two components:

4(x3)3⋅x3+5x323⋅x34 \sqrt[3]{(x^3)^3 \cdot x} + 5 x^3 \sqrt[3]{2^3 \cdot x}

Using the property of cube roots that states $\sqrt[3]{a^3} = a$, we can rewrite the expression as:

4x3x3+5x323⋅x34 x^3 \sqrt[3]{x} + 5 x^3 \sqrt[3]{2^3 \cdot x}

Factoring Out Common Terms

We can factor out the common term $x^3$ from both components:

x3(4x3+523⋅x3)x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{2^3 \cdot x})

Simplifying the Cube Root

We can simplify the cube root by using the property that states $\sqrt[3]{a^3} = a$. Since $2^3 = 8$, we can rewrite the expression as:

x3(4x3+58⋅x3)x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{8 \cdot x})

Simplifying the Expression Further

We can simplify the expression further by combining the two cube roots:

x3(4x3+583x3)x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{8} \sqrt[3]{x})

Using the Property of Cube Roots

We can use the property of cube roots that states $\sqrt[3]{a} \sqrt[3]{b} = \sqrt[3]{ab}$ to simplify the expression:

x3(4x3+58x3)x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{8x})

Simplifying the Cube Root

We can simplify the cube root by using the property that states $\sqrt[3]{a^3} = a$. Since $8 = 2^3$, we can rewrite the expression as:

x3(4x3+523x3)x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{2^3x})

Simplifying the Expression Further

We can simplify the expression further by combining the two cube roots:

x3(4x3+5233x3)x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{2^3} \sqrt[3]{x})

Using the Property of Cube Roots

We can use the property of cube roots that states $\sqrt[3]{a} \sqrt[3]{b} = \sqrt[3]{ab}$ to simplify the expression:

x3(4x3+58x3)x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{8x})

Simplifying the Cube Root

We can simplify the cube root by using the property that states $\sqrt[3]{a^3} = a$. Since $8 = 2^3$, we can rewrite the expression as:

x3(4x3+5⋅2x3)x^3 (4 \sqrt[3]{x} + 5 \cdot 2 \sqrt[3]{x})

Simplifying the Expression Further

We can simplify the expression further by combining the two terms:

x3(4x3+10x3)x^3 (4 \sqrt[3]{x} + 10 \sqrt[3]{x})

Factoring Out Common Terms

We can factor out the common term $\sqrt[3]{x}$ from both terms:

x3(4+10)x3x^3 (4 + 10) \sqrt[3]{x}

Simplifying the Expression

We can simplify the expression by combining the two terms:

x3(14)x3x^3 (14) \sqrt[3]{x}

Simplifying the Expression Further

We can simplify the expression further by combining the two terms:

14x3x314 x^3 \sqrt[3]{x}

Final Answer

The final answer is $14 x^3 \sqrt[3]{x}$.

Frequently Asked Questions

Q: What is the property of cube roots that states $\sqrt[3]{a^3} = a$?

A: This property states that the cube root of a number raised to the power of 3 is equal to the number itself. For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$.

Q: How can we simplify the expression $4 \sqrt[3]{x^{10}}+5 x^3 \sqrt[3]{8 x}$?

A: We can simplify the expression by breaking it down into its individual components and then simplifying each component separately. We can use the property of cube roots that states $\sqrt[3]{a^3} = a$ to simplify the expression.

Q: What is the simplified form of the expression $4 \sqrt[3]{x^{10}}+5 x^3 \sqrt[3]{8 x}$?

A: The simplified form of the expression is $14 x^3 \sqrt[3]{x}$.

Q: How can we factor out common terms from the expression $x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{8 \cdot x})$?

A: We can factor out the common term $x^3$ from both components of the expression.

Q: What is the property of cube roots that states $\sqrt[3]{a} \sqrt[3]{b} = \sqrt[3]{ab}$?

A: This property states that the cube root of the product of two numbers is equal to the product of their cube roots. For example, $\sqrt[3]{4} \sqrt[3]{9} = \sqrt[3]{36}$.

Q: How can we simplify the expression $x^3 (4 \sqrt[3]{x} + 5 \sqrt[3]{8x})$?

A: We can simplify the expression by using the property of cube roots that states $\sqrt[3]{a} \sqrt[3]{b} = \sqrt[3]{ab}$.

Q: What is the final answer to the problem?

A: The final answer is $14 x^3 \sqrt[3]{x}$.

Additional Resources

  • For more information on simplifying algebraic expressions with cube roots, see the article "Simplifying Algebraic Expressions with Cube Roots".
  • For more practice problems on simplifying algebraic expressions with cube roots, see the worksheet "Simplifying Algebraic Expressions with Cube Roots".

Conclusion

Simplifying algebraic expressions with cube roots can be a challenging task, but with the right techniques and properties, it can be done. By breaking down the expression into its individual components and then simplifying each component separately, we can simplify the expression and find the final answer.