Which Expression Is Equivalent To $2 \sqrt[3]{x^2} \cdot \sqrt{16 X}$, If $x\ \textgreater \ 0$?A. $8 X^6 \sqrt{x}$ B. $ 8 1 3 8 \sqrt[3]{1} 8 3 1 ​ [/tex] C. $8 X$ D. $2 X \sqrt[3]{16 X}$

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Understanding the Problem

When dealing with expressions involving radicals, it's essential to understand the properties of radicals and how they interact with each other. In this problem, we're given the expression $2 \sqrt[3]{x^2} \cdot \sqrt{16 x}$ and asked to find an equivalent expression, given that $x\ \textgreater \ 0$.

Breaking Down the Expression

To simplify the given expression, we can start by breaking it down into its components. We have $2 \sqrt[3]{x^2}$ and $\sqrt{16 x}$. We can simplify each of these components separately before combining them.

Simplifying the Cube Root

The cube root of $x^2$ can be simplified as follows:

x23=x23⋅x23⋅x23=x2/3\sqrt[3]{x^2} = \sqrt[3]{x^2} \cdot \sqrt[3]{x^2} \cdot \sqrt[3]{x^2} = x^{2/3}

Simplifying the Square Root

The square root of $16 x$ can be simplified as follows:

16x=16⋅x=4x\sqrt{16 x} = \sqrt{16} \cdot \sqrt{x} = 4 \sqrt{x}

Combining the Components

Now that we have simplified each component, we can combine them to get the final expression:

2x23⋅16x=2x2/3⋅4x2 \sqrt[3]{x^2} \cdot \sqrt{16 x} = 2 x^{2/3} \cdot 4 \sqrt{x}

Simplifying the Expression

To simplify the expression further, we can use the properties of exponents. We can rewrite $x^{2/3}$ as $(x{1/3})2$:

2x2/3⋅4x=2(x1/3)2⋅4x2 x^{2/3} \cdot 4 \sqrt{x} = 2 (x^{1/3})^2 \cdot 4 \sqrt{x}

Using the Power Rule

We can use the power rule to simplify the expression further:

2(x1/3)2⋅4x=2x2/3⋅4x1/22 (x^{1/3})^2 \cdot 4 \sqrt{x} = 2 x^{2/3} \cdot 4 x^{1/2}

Combining Like Terms

We can combine like terms to get the final expression:

2x2/3⋅4x1/2=8x2/3+1/22 x^{2/3} \cdot 4 x^{1/2} = 8 x^{2/3 + 1/2}

Simplifying the Exponent

We can simplify the exponent by finding a common denominator:

2/3+1/2=4/6+3/6=7/62/3 + 1/2 = 4/6 + 3/6 = 7/6

Final Expression

The final expression is:

8x7/68 x^{7/6}

Comparing with the Options

Now that we have simplified the expression, we can compare it with the options given:

A. $8 x^6 \sqrt{x}$ B. $8 \sqrt[3]{1}$ C. $8 x$ D. $2 x \sqrt[3]{16 x}$

Conclusion

Based on our simplification, we can see that the correct answer is:

A. $8 x^6 \sqrt{x}$

This is because $8 x^{7/6}$ is equivalent to $8 x^{6/6} \cdot x^{1/6}$, which simplifies to $8 x^6 \sqrt{x}$.

Final Answer

The final answer is A. $8 x^6 \sqrt{x}$.

Frequently Asked Questions

In the previous article, we explored the concept of radicals and exponents, and how to simplify expressions involving these mathematical operations. Here, we'll answer some frequently asked questions related to this topic.

Q: What is the difference between a radical and an exponent?

A: A radical is a mathematical operation that involves taking the square root or cube root of a number, while an exponent is a mathematical operation that involves raising a number to a power.

Q: How do I simplify an expression with a radical?

A: To simplify an expression with a radical, you can start by breaking it down into its components. Identify the radical and the number inside it, and then simplify the number inside the radical. If the number inside the radical can be simplified, you can rewrite it as a product of a perfect square and a remaining factor.

Q: What is the rule for multiplying radicals?

A: When multiplying radicals, you can multiply the numbers inside the radicals and then take the product of the radicals. For example, $\sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6}$.

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

A: To simplify an expression with a cube root, you can start by breaking it down into its components. Identify the cube root and the number inside it, and then simplify the number inside the cube root. If the number inside the cube root can be simplified, you can rewrite it as a product of a perfect cube and a remaining factor.

Q: What is the rule for dividing radicals?

A: When dividing radicals, you can divide the numbers inside the radicals and then take the quotient of the radicals. For example, $\sqrt{8} \div \sqrt{2} = \sqrt{8 \div 2} = \sqrt{4} = 2$.

Q: How do I simplify an expression with a combination of radicals and exponents?

A: To simplify an expression with a combination of radicals and exponents, you can start by simplifying the radicals and then simplifying the exponents. For example, $2 \sqrt{3} \cdot 3^2 = 2 \cdot 3 \cdot \sqrt{3} \cdot 3^2 = 6 \cdot 9 \sqrt{3} = 54 \sqrt{3}$.

Q: What is the rule for raising a radical to a power?

A: When raising a radical to a power, you can raise the number inside the radical to that power and then take the product of the radical and the power. For example, $(\sqrt{2})^3 = \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = 2 \sqrt{2}$.

Q: How do I simplify an expression with a combination of radicals and powers?

A: To simplify an expression with a combination of radicals and powers, you can start by simplifying the radicals and then simplifying the powers. For example, $2 \sqrt{3} \cdot 3^2 = 2 \cdot 3 \cdot \sqrt{3} \cdot 3^2 = 6 \cdot 9 \sqrt{3} = 54 \sqrt{3}$.

Q: What is the rule for simplifying a radical expression with a coefficient?

A: When simplifying a radical expression with a coefficient, you can simplify the coefficient and the radical separately. For example, $6 \sqrt{3} = 6 \cdot \sqrt{3} = 6 \cdot \sqrt{3}$.

Q: How do I simplify an expression with a combination of radicals and fractions?

A: To simplify an expression with a combination of radicals and fractions, you can start by simplifying the radicals and then simplifying the fractions. For example, $\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$.

Q: What is the rule for simplifying a radical expression with a variable?

A: When simplifying a radical expression with a variable, you can simplify the variable and the radical separately. For example, $\sqrt{x} = \sqrt{x}$.

Q: How do I simplify an expression with a combination of radicals and variables?

A: To simplify an expression with a combination of radicals and variables, you can start by simplifying the radicals and then simplifying the variables. For example, $\sqrt{x} \cdot \sqrt{y} = \sqrt{x \cdot y}$.

Q: What is the rule for simplifying a radical expression with a negative exponent?

A: When simplifying a radical expression with a negative exponent, you can simplify the exponent and the radical separately. For example, $\sqrt{x}^{-2} = \frac{1}{\sqrt{x}^2} = \frac{1}{x}$.

Q: How do I simplify an expression with a combination of radicals and negative exponents?

A: To simplify an expression with a combination of radicals and negative exponents, you can start by simplifying the radicals and then simplifying the negative exponents. For example, $\sqrt{x}^{-2} \cdot \sqrt{y} = \frac{\sqrt{y}}{x}$.

Q: What is the rule for simplifying a radical expression with a rational exponent?

A: When simplifying a radical expression with a rational exponent, you can simplify the exponent and the radical separately. For example, $\sqrt{x}^{3/2} = x^{3/2}$.

Q: How do I simplify an expression with a combination of radicals and rational exponents?

A: To simplify an expression with a combination of radicals and rational exponents, you can start by simplifying the radicals and then simplifying the rational exponents. For example, $\sqrt{x}^{3/2} \cdot \sqrt{y} = x^{3/2} \cdot \sqrt{y}$.

Q: What is the rule for simplifying a radical expression with a complex number?

A: When simplifying a radical expression with a complex number, you can simplify the complex number and the radical separately. For example, $\sqrt{3 + 4i} = \sqrt{3 + 4i}$.

Q: How do I simplify an expression with a combination of radicals and complex numbers?

A: To simplify an expression with a combination of radicals and complex numbers, you can start by simplifying the radicals and then simplifying the complex numbers. For example, $\sqrt{3 + 4i} \cdot \sqrt{2 + 3i} = \sqrt{(3 + 4i)(2 + 3i)}$.

Q: What is the rule for simplifying a radical expression with a matrix?

A: When simplifying a radical expression with a matrix, you can simplify the matrix and the radical separately. For example, $\sqrt{\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}} = \sqrt{\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}}$.

Q: How do I simplify an expression with a combination of radicals and matrices?

A: To simplify an expression with a combination of radicals and matrices, you can start by simplifying the radicals and then simplifying the matrices. For example, $\sqrt{\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}} \cdot \sqrt{\begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}} = \sqrt{\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}}$.

Q: What is the rule for simplifying a radical expression with a vector?

A: When simplifying a radical expression with a vector, you can simplify the vector and the radical separately. For example, $\sqrt{\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}} = \sqrt{\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}}$.

Q: How do I simplify an expression with a combination of radicals and vectors?

A: To simplify an expression with a combination of radicals and vectors, you can start by simplifying the radicals and then simplifying the vectors. For example, $\sqrt{\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}} \cdot \sqrt{\begin{bmatrix} 4 \ 5 \ 6 \end{bmatrix}} = \sqrt{\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} \cdot \begin{bmatrix} 4 \ 5 \ 6 \end{bmatrix}}$.

Q: What is the rule for simplifying a radical expression with a tensor?

A: When simplifying a radical expression with a tensor, you can simplify the tensor and the radical separately. For example, $\sqrt{\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}} = \sqrt{\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}}$.

Q: How do I simplify an expression with a combination of radicals and tensors?

A: To simplify an expression with a