If \[$ X \neq 0 \$\], What Is The Sum Of \[$ 4 \sqrt[3]{x^{10}} + 5 X^3 \sqrt[3]{8 X} \$\] In Simplest Form?
Understanding the Problem
The given problem involves simplifying an algebraic expression that contains cube roots and powers of a variable x. To simplify the expression, we need to apply the properties of exponents and cube roots.
Simplifying the Cube Roots
The expression contains two cube roots: { \sqrt[3]{x^{10}} $}$ and { \sqrt[3]{8 x} $}$. We can simplify these cube roots by applying the property { \sqrt[3]{a^n} = a^{\frac{n}{3}} $}$.
Simplifying the First Cube Root
Using the property, we can simplify the first cube root as follows:
{ \sqrt[3]{x^{10}} = x^{\frac{10}{3}} $}$
Simplifying the Second Cube Root
Similarly, we can simplify the second cube root as follows:
{ \sqrt[3]{8 x} = \sqrt[3]{2^3 x} = 2 x^{\frac{1}{3}} $}$
Substituting the Simplified Cube Roots
Now, we can substitute the simplified cube roots back into the original expression:
{ 4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x} = 4 x^{\frac{10}{3}} + 5 x^3 (2 x^{\frac{1}{3}}) $}$
Simplifying the Expression
We can simplify the expression further by applying the distributive property:
{ 4 x^{\frac{10}{3}} + 10 x^{\frac{10}{3}} $}$
Combining Like Terms
Now, we can combine the like terms:
{ 14 x^{\frac{10}{3}} $}$
Final Answer
Therefore, the sum of { 4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x} $}$ in simplest form is { 14 x^{\frac{10}{3}} $}$.
Conclusion
In this problem, we simplified an algebraic expression that contained cube roots and powers of a variable x. We applied the properties of exponents and cube roots to simplify the expression and arrived at the final answer.
Step-by-Step Solution
Here's a step-by-step solution to the problem:
- Simplify the first cube root: { \sqrt[3]{x^{10}} = x^{\frac{10}{3}} $}$
- Simplify the second cube root: { \sqrt[3]{8 x} = 2 x^{\frac{1}{3}} $}$
- Substitute the simplified cube roots back into the original expression: { 4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x} = 4 x^{\frac{10}{3}} + 5 x^3 (2 x^{\frac{1}{3}}) $}$
- Simplify the expression further by applying the distributive property: { 4 x^{\frac{10}{3}} + 10 x^{\frac{10}{3}} $}$
- Combine the like terms: { 14 x^{\frac{10}{3}} $}$
Frequently Asked Questions
- What is the sum of { 4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x} $}$ in simplest form?
- How do you simplify an algebraic expression that contains cube roots and powers of a variable x?
- What are the properties of exponents and cube roots that we need to apply to simplify the expression?
Related Topics
- Simplifying algebraic expressions
- Properties of exponents and cube roots
- Combining like terms
References
Frequently Asked Questions
Q: What is the sum of { 4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x} $}$ in simplest form?
A: The sum of { 4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x} $}$ in simplest form is { 14 x^{\frac{10}{3}} $}$.
Q: How do you simplify an algebraic expression that contains cube roots and powers of a variable x?
A: To simplify an algebraic expression that contains cube roots and powers of a variable x, you need to apply the properties of exponents and cube roots. Specifically, you can use the property { \sqrt[3]{a^n} = a^{\frac{n}{3}} $}$ to simplify the cube roots.
Q: What are the properties of exponents and cube roots that we need to apply to simplify the expression?
A: The properties of exponents and cube roots that we need to apply to simplify the expression are:
- { \sqrt[3]{a^n} = a^{\frac{n}{3}} $}$
- { a^m \cdot a^n = a^{m+n} $}$
- { a^m \cdot b^m = (ab)^m $}$
Q: How do you combine like terms in an algebraic expression?
A: To combine like terms in an algebraic expression, you need to identify the terms that have the same variable and exponent, and then add or subtract their coefficients.
Q: What is the final answer to the problem?
A: The final answer to the problem is { 14 x^{\frac{10}{3}} $}$.
Step-by-Step Solution
Here's a step-by-step solution to the problem:
- Simplify the first cube root: { \sqrt[3]{x^{10}} = x^{\frac{10}{3}} $}$
- Simplify the second cube root: { \sqrt[3]{8 x} = 2 x^{\frac{1}{3}} $}$
- Substitute the simplified cube roots back into the original expression: { 4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x} = 4 x^{\frac{10}{3}} + 5 x^3 (2 x^{\frac{1}{3}}) $}$
- Simplify the expression further by applying the distributive property: { 4 x^{\frac{10}{3}} + 10 x^{\frac{10}{3}} $}$
- Combine the like terms: { 14 x^{\frac{10}{3}} $}$
Related Topics
- Simplifying algebraic expressions
- Properties of exponents and cube roots
- Combining like terms