Simplify: $\sqrt[3]{y}\left(8 \sqrt[3]{y^2}+\sqrt[3]{y^5}-\sqrt[3]{27 Y^2}\right$\]A. $8 Y+y^2-3 Y$B. $8 Y+y \sqrt[3]{y^2}-3 \sqrt[3]{y^2}$C. $8 \sqrt[3]{y^3}+\sqrt[3]{y^6}-\sqrt[3]{27 Y^3}$D. $5 Y+y^2$

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Understanding the Expression

The given expression involves cube roots and a combination of terms within parentheses. To simplify this expression, we need to apply the properties of cube roots and perform the necessary operations.

Breaking Down the Expression

Let's start by breaking down the expression into smaller parts. We can rewrite the given expression as:

$\sqrt[3]{y}\left(8 \sqrt[3]{y^2}+\sqrt[3]{y^5}-\sqrt[3]{27 y^2}\right)$

Simplifying the Cube Roots

We can simplify the cube roots by expressing them in terms of powers of $y$. We know that $\sqrt[3]{y} = y^{\frac{1}{3}}$, $\sqrt[3]{y^2} = y^{\frac{2}{3}}$, and $\sqrt[3]{y^5} = y^{\frac{5}{3}}$.

Applying the Properties of Exponents

Using the properties of exponents, we can rewrite the expression as:

$y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}+y^{\frac{5}{3}}-y^{\frac{4}{3}}\right)$

Simplifying the Expression

Now, we can simplify the expression by combining like terms. We can rewrite the expression as:

$y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}+y^{\frac{5}{3}}-y^{\frac{4}{3}}\right)$

$= y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}+y^{\frac{2}{3}}y^{\frac{3}{3}}-y^{\frac{2}{3}}y^{\frac{2}{3}}\right)$

$= y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}+y^{\frac{2}{3}}y- y^{\frac{2}{3}}y\right)$

$= y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}\right)$

$= 8 y^{\frac{3}{3}}$

Final Answer

The final answer is $8 y$.

Explanation

The correct answer is $8 y$. This is because the cube root of $y^3$ is $y$, and the cube root of $y^2$ is $y^{\frac{2}{3}}$. Therefore, the expression simplifies to $8 y$.

Comparison with Options

Let's compare the final answer with the options provided:

A. $8 y+y^2-3 y$ B. $8 y+y \sqrt[3]{y^2}-3 \sqrt[3]{y^2}$ C. $8 \sqrt[3]{y^3}+\sqrt[3]{y^6}-\sqrt[3]{27 y^3}$ D. $5 y+y^2$

The final answer $8 y$ matches option A.

Conclusion

In conclusion, the correct answer is $8 y$. This is because the cube root of $y^3$ is $y$, and the cube root of $y^2$ is $y^{\frac{2}{3}}$. Therefore, the expression simplifies to $8 y$.

Final Answer

The final answer is $8 y$.

Discussion

This problem involves simplifying an expression that contains cube roots and a combination of terms within parentheses. To simplify this expression, we need to apply the properties of cube roots and perform the necessary operations.

Key Concepts

• Cube roots
• Properties of exponents
• Simplifying expressions

Related Problems

• Simplifying expressions with cube roots
• Applying the properties of exponents
• Simplifying expressions with a combination of terms within parentheses

Practice Problems

• Simplify the expression: $\sqrt[3]{y}\left(4 \sqrt[3]{y^2}+\sqrt[3]{y^4}-\sqrt[3]{81 y^2}\right)$
• Simplify the expression: $\sqrt[3]{y}\left(6 \sqrt[3]{y^2}+\sqrt[3]{y^5}-\sqrt[3]{64 y^2}\right)$

Solutions

• The final answer is $4 y+y^2-3 y$.
• The final answer is $6 y+y \sqrt[3]{y^2}-3 \sqrt[3]{y^2}$.
  

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Frequently Asked Questions

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to break it down into smaller parts. We can rewrite the expression as:

$\sqrt[3]{y}\left(8 \sqrt[3]{y^2}+\sqrt[3]{y^5}-\sqrt[3]{27 y^2}\right)$

Q: How do we simplify the cube roots in the expression?

A: We can simplify the cube roots by expressing them in terms of powers of $y$. We know that $\sqrt[3]{y} = y^{\frac{1}{3}}$, $\sqrt[3]{y^2} = y^{\frac{2}{3}}$, and $\sqrt[3]{y^5} = y^{\frac{5}{3}}$.

Q: What is the next step in simplifying the expression?

A: The next step in simplifying the expression is to apply the properties of exponents. We can rewrite the expression as:

$y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}+y^{\frac{5}{3}}-y^{\frac{4}{3}}\right)$

Q: How do we simplify the expression further?

A: We can simplify the expression further by combining like terms. We can rewrite the expression as:

$y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}+y^{\frac{5}{3}}-y^{\frac{4}{3}}\right)$

$= y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}+y^{\frac{2}{3}}y^{\frac{3}{3}}-y^{\frac{2}{3}}y^{\frac{2}{3}}\right)$

$= y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}+y^{\frac{2}{3}}y- y^{\frac{2}{3}}y\right)$

$= y^{\frac{1}{3}}\left(8 y^{\frac{2}{3}}\right)$

$= 8 y^{\frac{3}{3}}$

Q: What is the final answer?

A: The final answer is $8 y$.

Q: Why is the final answer $8 y$?

A: The final answer is $8 y$ because the cube root of $y^3$ is $y$, and the cube root of $y^2$ is $y^{\frac{2}{3}}$. Therefore, the expression simplifies to $8 y$.

Q: How does the final answer compare to the options provided?

A: The final answer $8 y$ matches option A.

Q: What are some related problems that involve simplifying expressions with cube roots?

A: Some related problems that involve simplifying expressions with cube roots include:

• Simplifying the expression: $\sqrt[3]{y}\left(4 \sqrt[3]{y^2}+\sqrt[3]{y^4}-\sqrt[3]{81 y^2}\right)$
• Simplifying the expression: $\sqrt[3]{y}\left(6 \sqrt[3]{y^2}+\sqrt[3]{y^5}-\sqrt[3]{64 y^2}\right)$

Q: What are some key concepts that are involved in simplifying expressions with cube roots?

A: Some key concepts that are involved in simplifying expressions with cube roots include:

• Cube roots
• Properties of exponents
• Simplifying expressions

Q: What are some practice problems that involve simplifying expressions with cube roots?

A: Some practice problems that involve simplifying expressions with cube roots include:

• Simplifying the expression: $\sqrt[3]{y}\left(2 \sqrt[3]{y^2}+\sqrt[3]{y^3}-\sqrt[3]{27 y^2}\right)$
• Simplifying the expression: $\sqrt[3]{y}\left(3 \sqrt[3]{y^2}+\sqrt[3]{y^4}-\sqrt[3]{64 y^2}\right)$

Conclusion

In conclusion, simplifying expressions with cube roots involves applying the properties of exponents and combining like terms. The final answer is $8 y$, and it matches option A. Some related problems that involve simplifying expressions with cube roots include simplifying the expression: $\sqrt[3]{y}\left(4 \sqrt[3]{y^2}+\sqrt[3]{y^4}-\sqrt[3]{81 y^2}\right)$ and simplifying the expression: $\sqrt[3]{y}\left(6 \sqrt[3]{y^2}+\sqrt[3]{y^5}-\sqrt[3]{64 y^2}\right)$.