Simplify: 27 X 7 3 \sqrt[3]{27 X^7} 3 27 X 7 ​ A. 3 X 4 3 X^4 3 X 4 B. 3 X X 2 3 3 X \sqrt[3]{x^2} 3 X 3 X 2 ​ C. 9 X 2 X 3 9 X^2 \sqrt[3]{x} 9 X 2 3 X ​ D. 3 X 2 X 3 3 X^2 \sqrt[3]{x} 3 X 2 3 X ​

Understanding the Problem

To simplify the given expression, we need to apply the properties of radicals and exponents. The expression $\sqrt[3]{27 x^7}$ involves a cube root, which means we are looking for a value that, when multiplied by itself three times, gives us the original expression. We will use the properties of radicals and exponents to simplify this expression.

Breaking Down the Expression

The expression $\sqrt[3]{27 x^7}$ can be broken down into two parts: $\sqrt[3]{27}$ and $\sqrt[3]{x^7}$. We can simplify each part separately.

Simplifying the Radical Part

The number 27 can be expressed as $3^3$, which means that $\sqrt[3]{27} = \sqrt[3]{3^3} = 3$. Therefore, the radical part of the expression simplifies to 3.

Simplifying the Exponential Part

The expression $\sqrt[3]{x^7}$ can be simplified using the property of exponents that states $\sqrt[n]{x^n} = x$. In this case, we have $\sqrt[3]{x^7} = x^{\frac{7}{3}}$. However, we can further simplify this expression by expressing $x^7$ as $x^6 \cdot x$. This gives us $\sqrt[3]{x^7} = \sqrt[3]{x^6 \cdot x} = x^2 \cdot \sqrt[3]{x}$.

Combining the Simplified Parts

Now that we have simplified the radical and exponential parts, we can combine them to get the final simplified expression. We have $\sqrt[3]{27 x^7} = 3 \cdot x^2 \cdot \sqrt[3]{x} = 3 x^2 \sqrt[3]{x}$.

Conclusion

In conclusion, the simplified expression for $\sqrt[3]{27 x^7}$ is $3 x^2 \sqrt[3]{x}$. This expression involves a cube root and an exponential term, and we used the properties of radicals and exponents to simplify it.

Answer

The correct answer is D. $3 x^2 \sqrt[3]{x}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Break down the expression $\sqrt[3]{27 x^7}$ into two parts: $\sqrt[3]{27}$ and $\sqrt[3]{x^7}$.
2. Simplify the radical part: $\sqrt[3]{27} = \sqrt[3]{3^3} = 3$.
3. Simplify the exponential part: $\sqrt[3]{x^7} = \sqrt[3]{x^6 \cdot x} = x^2 \cdot \sqrt[3]{x}$.
4. Combine the simplified parts: $\sqrt[3]{27 x^7} = 3 \cdot x^2 \cdot \sqrt[3]{x} = 3 x^2 \sqrt[3]{x}$.

Final Answer

The final answer is D. $3 x^2 \sqrt[3]{x}$.

Understanding the Problem

To simplify the given expression, we need to apply the properties of radicals and exponents. The expression $\sqrt[3]{27 x^7}$ involves a cube root, which means we are looking for a value that, when multiplied by itself three times, gives us the original expression. We will use the properties of radicals and exponents to simplify this expression.

Q&A

Q: What is the cube root of 27?

A: The cube root of 27 is 3, because $3^3 = 27$.

Q: How do we simplify the expression $\sqrt[3]{x^7}$?

A: We can simplify the expression $\sqrt[3]{x^7}$ by expressing $x^7$ as $x^6 \cdot x$. This gives us $\sqrt[3]{x^7} = \sqrt[3]{x^6 \cdot x} = x^2 \cdot \sqrt[3]{x}$.

Q: How do we combine the simplified parts?

A: We can combine the simplified parts by multiplying them together. This gives us $\sqrt[3]{27 x^7} = 3 \cdot x^2 \cdot \sqrt[3]{x} = 3 x^2 \sqrt[3]{x}$.

Q: What is the final simplified expression?

A: The final simplified expression is $3 x^2 \sqrt[3]{x}$.

Q: Why is this expression simplified?

A: This expression is simplified because we have applied the properties of radicals and exponents to break down the original expression into smaller parts, and then combined those parts to get the final simplified expression.

Q: What is the correct answer?

A: The correct answer is D. $3 x^2 \sqrt[3]{x}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Break down the expression $\sqrt[3]{27 x^7}$ into two parts: $\sqrt[3]{27}$ and $\sqrt[3]{x^7}$.
2. Simplify the radical part: $\sqrt[3]{27} = \sqrt[3]{3^3} = 3$.
3. Simplify the exponential part: $\sqrt[3]{x^7} = \sqrt[3]{x^6 \cdot x} = x^2 \cdot \sqrt[3]{x}$.
4. Combine the simplified parts: $\sqrt[3]{27 x^7} = 3 \cdot x^2 \cdot \sqrt[3]{x} = 3 x^2 \sqrt[3]{x}$.

Final Answer

The final answer is D. $3 x^2 \sqrt[3]{x}$.

Common Mistakes

• Not breaking down the expression into smaller parts
• Not simplifying the radical and exponential parts correctly
• Not combining the simplified parts correctly

Tips and Tricks

• Make sure to break down the expression into smaller parts
• Simplify the radical and exponential parts correctly
• Combine the simplified parts correctly

Conclusion

In conclusion, the simplified expression for $\sqrt[3]{27 x^7}$ is $3 x^2 \sqrt[3]{x}$. This expression involves a cube root and an exponential term, and we used the properties of radicals and exponents to simplify it.