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

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression: $4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8x}$. We will break down the expression into manageable parts, apply the necessary rules and formulas, and arrive at the simplest form.

Understanding Radical Expressions

Before we dive into the simplification process, let's take a moment to understand what radical expressions are. A radical expression is a mathematical expression that contains a root or a power of a number. In this case, we have expressions with cube roots, denoted by the symbol $\sqrt[3]{x}$.

Breaking Down the Expression

To simplify the given expression, we need to break it down into smaller parts. Let's start by identifying the individual terms:

  • 4x1034 \sqrt[3]{x^{10}}

  • 5x38x35 x^3 \sqrt[3]{8x}

Simplifying the First Term

The first term is $4 \sqrt[3]{x^{10}}$. To simplify this term, we need to apply the rule for simplifying cube roots:

xn3=xn3\sqrt[3]{x^n} = x^{\frac{n}{3}}

Using this rule, we can rewrite the first term as:

4x103=4x1034 \sqrt[3]{x^{10}} = 4 x^{\frac{10}{3}}

Simplifying the Second Term

The second term is $5 x^3 \sqrt[3]{8x}$. To simplify this term, we need to apply the rule for simplifying cube roots:

xn3=xn3\sqrt[3]{x^n} = x^{\frac{n}{3}}

Using this rule, we can rewrite the second term as:

5x38x3=5x3x33835 x^3 \sqrt[3]{8x} = 5 x^3 x^{\frac{3}{3}} \sqrt[3]{8}

Simplifying further, we get:

5x3x3383=5x4835 x^3 x^{\frac{3}{3}} \sqrt[3]{8} = 5 x^4 \sqrt[3]{8}

Simplifying the Cube Root

The cube root of 8 can be simplified as:

83=233=2\sqrt[3]{8} = \sqrt[3]{2^3} = 2

Using this simplification, we can rewrite the second term as:

5x483=5x4⋅2=10x45 x^4 \sqrt[3]{8} = 5 x^4 \cdot 2 = 10 x^4

Combining the Terms

Now that we have simplified both terms, we can combine them to get the final expression:

4x103+10x44 x^{\frac{10}{3}} + 10 x^4

Simplifying the Expression

To simplify the expression further, we need to find a common denominator for the two terms. The common denominator is $x^{\frac{10}{3}}$.

Using this common denominator, we can rewrite the expression as:

4x103+10x4=4x103+10x103⋅x73x734 x^{\frac{10}{3}} + 10 x^4 = 4 x^{\frac{10}{3}} + \frac{10 x^{\frac{10}{3}} \cdot x^{\frac{7}{3}}}{x^{\frac{7}{3}}}

Simplifying further, we get:

4x103+10x103⋅x73x73=4x103+10x1734 x^{\frac{10}{3}} + \frac{10 x^{\frac{10}{3}} \cdot x^{\frac{7}{3}}}{x^{\frac{7}{3}}} = 4 x^{\frac{10}{3}} + 10 x^{\frac{17}{3}}

Conclusion

In this article, we have simplified the given radical expression $4 \sqrt[3]x^{10}} + 5 x^3 \sqrt[3]{8x}$ in simplest form. We broke down the expression into manageable parts, applied the necessary rules and formulas, and arrived at the final expression $4 x^{\frac{10{3}} + 10 x^{\frac{17}{3}}$.

Final Answer

Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression: $4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8x}$. We broke down the expression into manageable parts, applied the necessary rules and formulas, and arrived at the simplest form. In this article, we will answer some frequently asked questions about simplifying radical expressions.

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

A: A radical expression is a mathematical expression that contains a root or a power of a number, whereas a polynomial expression is a mathematical expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication.

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

A: To simplify a radical expression with multiple terms, you need to break down the expression into smaller parts, apply the necessary rules and formulas, and then combine the simplified terms.

Q: What is the rule for simplifying cube roots?

A: The rule for simplifying cube roots is:

xn3=xn3\sqrt[3]{x^n} = x^{\frac{n}{3}}

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 apply the rule for simplifying cube roots and then simplify the resulting expression.

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

A: Yes, you can simplify a radical expression with a negative exponent by applying the rule for simplifying negative exponents:

x−n=1xnx^{-n} = \frac{1}{x^n}

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

A: To simplify a radical expression with a fraction in the radicand, you need to apply the rule for simplifying fractions:

xy3=x3y3\sqrt[3]{\frac{x}{y}} = \frac{\sqrt[3]{x}}{\sqrt[3]{y}}

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

A: No, you cannot simplify a radical expression with a decimal in the radicand. The radicand must be a whole number or a variable.

Q: How do I check if a radical expression is simplified?

A: To check if a radical expression is simplified, you need to ensure that the radicand is a whole number or a variable, and that the expression cannot be simplified further.

Conclusion

In this article, we have answered some frequently asked questions about simplifying radical expressions. We have covered topics such as the difference between radical expressions and polynomial expressions, simplifying radical expressions with multiple terms, and simplifying radical expressions with variables, fractions, and decimals in the radicand.

Final Tips

  • Always apply the necessary rules and formulas when simplifying radical expressions.
  • Break down the expression into smaller parts to make it easier to simplify.
  • Check if the radicand is a whole number or a variable before simplifying the expression.
  • Use the rule for simplifying cube roots to simplify radical expressions with cube roots.

Final Answer

The final answer is: $4 x^{\frac{10}{3}} + 10 x^{\frac{17}{3}}$