Rewrite The Expression In The Form K ⋅ Y N K \cdot Y^n K ⋅ Y N . Write The Exponent As An Integer, Fraction, Or An Exact Decimal (not A Mixed Number). 3 Y − 4 3 ⋅ 2 Y 3 = □ 3 Y^{-\frac{4}{3}} \cdot 2 \sqrt[3]{y} = \square 3 Y − 3 4 ​ ⋅ 2 3 Y ​ = □

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Understanding the Problem

In mathematics, we often encounter expressions that can be rewritten in a more simplified form. One such form is the expression $k \cdot y^n$, where $k$ is a constant and $n$ is an exponent. In this article, we will focus on rewriting the expression $3 y^{-\frac{4}{3}} \cdot 2 \sqrt[3]{y}$ in the form $k \cdot y^n$.

Rewriting Negative Exponents

The first step in rewriting the expression is to deal with the negative exponent. A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. In this case, we have $y^{-\frac{4}{3}}$, which can be rewritten as $\frac{1}{y^{\frac{4}{3}}}$.

import sympy as sp
y = sp.symbols('y')

expr = 3 * (1 / (y ** (4/3))) * 2 * (y ** (1/3))

Rewriting Radicals

The next step is to deal with the radical. A radical can be rewritten as a power by using the property $\sqrt[n]{x} = x^{\frac{1}{n}}$. In this case, we have $\sqrt[3]{y}$, which can be rewritten as $y^{\frac{1}{3}}$.

# Rewrite the radical
expr = expr.subs(sp.sqrt(y ** (1/3)), y ** (1/3))

Combining the Terms

Now that we have dealt with the negative exponent and the radical, we can combine the terms. We have $\frac{3}{y^{\frac{4}{3}}} \cdot 2 \cdot y^{\frac{1}{3}}$, which can be rewritten as $\frac{6}{y^{\frac{4}{3}}} \cdot y^{\frac{1}{3}}$.

# Combine the terms
expr = (3 * 2) / (y ** (4/3)) * y ** (1/3)

Simplifying the Expression

The final step is to simplify the expression. We can do this by combining the terms with the same base. In this case, we have $\frac{6}{y^{\frac{4}{3}}} \cdot y^{\frac{1}{3}}$, which can be rewritten as $\frac{6}{y^{\frac{4}{3} - \frac{1}{3}}} = \frac{6}{y^{\frac{3}{3}}} = \frac{6}{y}$.

# Simplify the expression
expr = 6 / y

Conclusion

In this article, we have rewritten the expression $3 y^{-\frac{4}{3}} \cdot 2 \sqrt[3]{y}$ in the form $k \cdot y^n$. We have dealt with the negative exponent and the radical, combined the terms, and simplified the expression. The final result is $\frac{6}{y}$.

Final Answer

The final answer is $\boxed{\frac{6}{y}}$.

Discussion

This problem is a great example of how to rewrite expressions in the form $k \cdot y^n$. It requires a good understanding of negative exponents, radicals, and simplifying expressions. If you have any questions or need further clarification, please don't hesitate to ask.

Related Problems

• Rewrite the expression $2 x^{-\frac{2}{3}} \cdot 3 \sqrt[3]{x}$ in the form $k \cdot x^n$.
• Rewrite the expression $4 y^{-\frac{5}{2}} \cdot 2 \sqrt[2]{y}$ in the form $k \cdot y^n$.
• Rewrite the expression $3 z^{-\frac{3}{4}} \cdot 2 \sqrt[4]{z}$ in the form $k \cdot z^n$.

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about rewriting expressions in the form $k \cdot y^n$.

Q: What is the form $k \cdot y^n$?

A: The form $k \cdot y^n$ is a way of writing an expression where $k$ is a constant and $n$ is an exponent. This form is useful for simplifying expressions and making them easier to work with.

Q: How do I rewrite an expression in the form $k \cdot y^n$?

A: To rewrite an expression in the form $k \cdot y^n$, you need to follow these steps:

1. Deal with any negative exponents by taking the reciprocal of the base.
2. Rewrite any radicals as powers by using the property $\sqrt[n]{x} = x^{\frac{1}{n}}$.
3. Combine the terms with the same base.
4. Simplify the expression by combining the terms with the same base.

Q: What is a negative exponent?

A: A negative exponent is a way of writing an expression where the exponent is a negative number. For example, $y^{-\frac{4}{3}}$ is a negative exponent.

Q: How do I deal with a negative exponent?

A: To deal with a negative exponent, you need to take the reciprocal of the base. For example, $y^{-\frac{4}{3}}$ can be rewritten as $\frac{1}{y^{\frac{4}{3}}}$.

Q: What is a radical?

A: A radical is a way of writing an expression where the base is raised to a fractional exponent. For example, $\sqrt[3]{y}$ is a radical.

Q: How do I rewrite a radical as a power?

A: To rewrite a radical as a power, you need to use the property $\sqrt[n]{x} = x^{\frac{1}{n}}$. For example, $\sqrt[3]{y}$ can be rewritten as $y^{\frac{1}{3}}$.

Q: Can I simplify an expression by combining the terms with the same base?

A: Yes, you can simplify an expression by combining the terms with the same base. For example, $\frac{6}{y^{\frac{4}{3}}} \cdot y^{\frac{1}{3}}$ can be simplified to $\frac{6}{y}$.

Q: What are some common mistakes to avoid when rewriting expressions in the form $k \cdot y^n$?

A: Some common mistakes to avoid when rewriting expressions in the form $k \cdot y^n$ include:

• Not dealing with negative exponents correctly
• Not rewriting radicals as powers correctly
• Not combining the terms with the same base correctly
• Not simplifying the expression correctly

Q: How can I practice rewriting expressions in the form $k \cdot y^n$?

A: You can practice rewriting expressions in the form $k \cdot y^n$ by working through examples and exercises. You can also try rewriting expressions in different forms, such as $k \cdot x^n$ or $k \cdot z^n$.

Conclusion

In this article, we have answered some of the most frequently asked questions about rewriting expressions in the form $k \cdot y^n$. We have covered topics such as negative exponents, radicals, and simplifying expressions. We hope this article has been helpful in understanding how to rewrite expressions in the form $k \cdot y^n$. If you have any questions or need further clarification, please don't hesitate to ask.

Final Answer

The final answer is $\boxed{\frac{6}{y}}$.

Discussion

This problem is a great example of how to rewrite expressions in the form $k \cdot y^n$. It requires a good understanding of negative exponents, radicals, and simplifying expressions. If you have any questions or need further clarification, please don't hesitate to ask.

Related Problems

• Rewrite the expression $2 x^{-\frac{2}{3}} \cdot 3 \sqrt[3]{x}$ in the form $k \cdot x^n$.
• Rewrite the expression $4 y^{-\frac{5}{2}} \cdot 2 \sqrt[2]{y}$ in the form $k \cdot y^n$.
• Rewrite the expression $3 z^{-\frac{3}{4}} \cdot 2 \sqrt[4]{z}$ in the form $k \cdot z^n$.

I hope this article has been helpful in understanding how to rewrite expressions in the form $k \cdot y^n$. If you have any questions or need further clarification, please don't hesitate to ask.