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

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 product of a constant and a cube root of a variable. The expression is $5(\sqrt[3]{x}) + 9(\sqrt[3]{x})$. 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 using the distributive property. This allows us to add the coefficients of the two terms, which are 5 and 9, respectively. The expression becomes:

$5(\sqrt[3]{x}) + 9(\sqrt[3]{x}) = (5 + 9)(\sqrt[3]{x})$

Applying the Distributive Property

Now, we can apply the distributive property to simplify the expression further. The distributive property states that for any numbers a, b, and c, the following equation holds:

$a(b + c) = ab + ac$

In this case, we have:

$(5 + 9)(\sqrt[3]{x}) = 14(\sqrt[3]{x})$

Simplifying the Cube Root

The cube root of a variable can be simplified by raising the variable to the power of 1/3. In this case, we have:

$\sqrt[3]{x} = x^{1/3}$

However, this is not necessary in this case, as the cube root is already in its simplest form.

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options to determine the correct answer. The options are:

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

Option A

Option A is $14(\sqrt[6]{x})$. To evaluate this option, we need to simplify the cube root of x to the sixth power. This can be done by raising the cube root to the power of 2, which is equivalent to raising x to the power of 2/3. However, this is not necessary in this case, as the cube root is already in its simplest form.

Option B

Option B is $14\left(\sqrt[6]{x^2}\right)$. To evaluate this option, we need to simplify the cube root of x squared to the sixth power. This can be done by raising the cube root to the power of 2, which is equivalent to raising x squared to the power of 2/3. However, this is not necessary in this case, as the cube root is already in its simplest form.

Option C

Option C is $14(\sqrt[3]{x})$. This option is identical to the simplified expression we obtained earlier.

Option D

Option D is $14\left(\sqrt[3]{x^2}\right)$. To evaluate this option, we need to simplify the cube root of x squared. This can be done by raising x squared to the power of 1/3.

Conclusion

Based on our analysis, we can conclude that the correct answer is:

C. $14(\sqrt[3]{x})$

This option is identical to the simplified expression we obtained earlier. The other options are incorrect because they either simplify the cube root to a different power or introduce unnecessary complexity.

Final Answer

The final answer is C. $14(\sqrt[3]{x})$.

Q: What is the distributive property in algebra?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by a constant or another expression. It states that for any numbers a, b, and c, the following equation holds:

$a(b + c) = ab + ac$

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

A: To simplify an algebraic expression with cube roots, you can start by combining like terms using the distributive property. Then, you can simplify the cube root by raising the variable to the power of 1/3. However, in many cases, the cube root is already in its simplest form, and no further simplification is necessary.

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

A: A cube root is a root that is raised to the power of 1/3, while a sixth root is a root that is raised to the power of 1/6. In other words, a cube root is a root that is three times more powerful than a sixth root.

Q: How do I evaluate an option that involves simplifying a cube root to a different power?

A: To evaluate an option that involves simplifying a cube root to a different power, you need to raise the cube root to the power of the new exponent. For example, if you have a cube root of x and you want to simplify it to the sixth power, you would raise the cube root to the power of 2, which is equivalent to raising x to the power of 2/3.

Q: What is the correct order of operations when simplifying an algebraic expression?

A: The correct order of operations when simplifying an algebraic expression is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponents next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I determine the correct answer from a list of options?

A: To determine the correct answer from a list of options, you need to carefully evaluate each option and compare it to the simplified expression you obtained earlier. Look for any differences in the expression, such as changes in the exponent or the variable. If you find any differences, you can eliminate that option as a possible answer.

Q: What if I'm still unsure about the correct answer?

A: If you're still unsure about the correct answer, you can try simplifying the expression further or using a different method to evaluate the options. You can also ask for help from a teacher or tutor if you're struggling with the concept.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through practice problems or exercises. You can also try simplifying expressions on your own and then checking your work against the correct answer. Additionally, you can use online resources or math software to help you practice and improve your skills.

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

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

• Forgetting to distribute the constant or expression to each term inside the parentheses
• Simplifying the expression incorrectly or making a mistake in the exponent
• Failing to evaluate expressions inside parentheses first
• Not following the correct order of operations

By avoiding these common mistakes and practicing regularly, you can improve your skills and become more confident when simplifying algebraic expressions.