What Is The Following Sum?$\[ 5(\sqrt[3]{x}) + 9(\sqrt[3]{x}) \\]A. \[$ 14(\sqrt[6]{x}) \$\] B. \[$ 14\left(\sqrt[6]{x^2}\right) \$\] C. \[$ 14(\sqrt[3]{x}) \$\] D. \[$ 14\left(\sqrt[3]{x^2}\right) \$\]

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Understanding the Problem

The given problem involves simplifying an algebraic expression that contains cube roots. We are asked to find the sum of two terms, each of which is a multiple of the cube root of x. The expression is: 5(x3)+9(x3)5(\sqrt[3]{x}) + 9(\sqrt[3]{x}). Our goal is to simplify this expression and determine the correct answer from the given options.

Simplifying the Expression

To simplify the expression, we can start by combining the two terms. Since both terms have the same cube root of x, we can add their coefficients (5 and 9) and keep the cube root of x as it is. This gives us: 5(x3)+9(x3)=(5+9)(x3)=14(x3)5(\sqrt[3]{x}) + 9(\sqrt[3]{x}) = (5 + 9)(\sqrt[3]{x}) = 14(\sqrt[3]{x}).

Analyzing the Options

Now that we have simplified the expression, let's analyze the given options:

A. 14(x6)14(\sqrt[6]{x}) B. 14(x26)14\left(\sqrt[6]{x^2}\right) C. 14(x3)14(\sqrt[3]{x}) D. 14(x23)14\left(\sqrt[3]{x^2}\right)

Evaluating Option A

Option A is 14(x6)14(\sqrt[6]{x}). To evaluate this option, we need to understand that x6\sqrt[6]{x} is equivalent to (x3)2(\sqrt[3]{x})^2. However, our simplified expression is 14(x3)14(\sqrt[3]{x}), not 14((x3)2)14((\sqrt[3]{x})^2). Therefore, option A is not correct.

Evaluating Option B

Option B is 14(x26)14\left(\sqrt[6]{x^2}\right). To evaluate this option, we need to understand that x26\sqrt[6]{x^2} is equivalent to (x23)2(\sqrt[3]{x^2})^2. However, our simplified expression is 14(x3)14(\sqrt[3]{x}), not 14(x23)14(\sqrt[3]{x^2}). Therefore, option B is not correct.

Evaluating Option C

Option C is 14(x3)14(\sqrt[3]{x}). As we simplified the expression earlier, this option matches our result. Therefore, option C is correct.

Evaluating Option D

Option D is 14(x23)14\left(\sqrt[3]{x^2}\right). To evaluate this option, we need to understand that x23\sqrt[3]{x^2} is equivalent to (x3)2(\sqrt[3]{x})^2. However, our simplified expression is 14(x3)14(\sqrt[3]{x}), not 14((x3)2)14((\sqrt[3]{x})^2). Therefore, option D is not correct.

Conclusion

In conclusion, the correct answer is option C, which is 14(x3)14(\sqrt[3]{x}). This is because our simplified expression matches this option exactly.

Frequently Asked Questions

Q: What is the cube root of x?

A: The cube root of x is a number that, when multiplied by itself twice, gives x. It is denoted by x3\sqrt[3]{x}.

Q: What is the sixth root of x?

A: The sixth root of x is a number that, when multiplied by itself five times, gives x. It is denoted by x6\sqrt[6]{x}.

Q: How do I simplify an algebraic expression that contains cube roots?

A: To simplify an algebraic expression that contains cube roots, you can start by combining like terms and then simplify the resulting expression.

Q: What is the difference between the cube root and the sixth root?

A: The cube root and the sixth root are both roots of a number, but they are different. The cube root is a number that, when multiplied by itself twice, gives the original number, while the sixth root is a number that, when multiplied by itself five times, gives the original number.

Final Answer

The final answer is option C, which is 14(x3)14(\sqrt[3]{x}).

Understanding Algebraic Expressions with Cube Roots

Algebraic expressions with cube roots are a fundamental concept in mathematics. They involve numbers and variables that are raised to the power of 1/3, denoted by the radical sign (√[3]). In this article, we will answer some frequently asked questions about algebraic expressions with cube roots.

Q&A Session

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself twice, gives the original number. It is denoted by √[3]x, where x is the number.

Q: How do I simplify an algebraic expression with a cube root?

A: To simplify an algebraic expression with a cube root, you can start by combining like terms and then simplify the resulting expression. For example, if you have the expression 2√[3]x + 3√[3]x, you can combine the like terms to get 5√[3]x.

Q: What is the difference between the cube root and the sixth root?

A: The cube root and the sixth root are both roots of a number, but they are different. The cube root is a number that, when multiplied by itself twice, gives the original number, while the sixth root is a number that, when multiplied by itself five times, gives the original number.

Q: How do I evaluate an expression with a cube root?

A: To evaluate an expression with a cube root, you need to follow the order of operations (PEMDAS). First, evaluate any expressions inside the parentheses, then evaluate any exponential expressions, and finally evaluate any multiplication and division operations.

Q: Can I simplify an expression with a cube root by multiplying it by a fraction?

A: Yes, you can simplify an expression with a cube root by multiplying it by a fraction. For example, if you have the expression 2√[3]x, you can multiply it by 1/2 to get √[3]x.

Q: How do I solve an equation with a cube root?

A: To solve an equation with a cube root, you need to isolate the cube root term on one side of the equation. Then, you can cube both sides of the equation to eliminate the cube root.

Q: What is the cube root of a negative number?

A: The cube root of a negative number is a negative number. For example, the cube root of -64 is -4.

Q: Can I simplify an expression with a cube root by using the product rule?

A: Yes, you can simplify an expression with a cube root by using the product rule. For example, if you have the expression 2√[3]x + 3√[3]x, you can use the product rule to combine the like terms.

Q: How do I find the cube root of a decimal number?

A: To find the cube root of a decimal number, you can use a calculator or a computer program. Alternatively, you can use a mathematical formula to estimate the cube root.

Conclusion

In conclusion, algebraic expressions with cube roots are a fundamental concept in mathematics. By understanding the properties of cube roots and how to simplify expressions with them, you can solve a wide range of mathematical problems. We hope that this article has been helpful in answering your questions about algebraic expressions with cube roots.

Final Tips

  • Always follow the order of operations (PEMDAS) when evaluating expressions with cube roots.
  • Use the product rule to simplify expressions with like terms.
  • Be careful when simplifying expressions with cube roots, as the cube root of a negative number is a negative number.
  • Use a calculator or a computer program to find the cube root of a decimal number.

Additional Resources

  • Khan Academy: Algebraic Expressions with Cube Roots
  • Mathway: Algebraic Expressions with Cube Roots
  • Wolfram Alpha: Algebraic Expressions with Cube Roots

Final Answer

The final answer is that algebraic expressions with cube roots are a fundamental concept in mathematics, and by understanding the properties of cube roots and how to simplify expressions with them, you can solve a wide range of mathematical problems.