What Is The Simplest Form Of X 4 Y 7 X 10 Y 4 3 \frac{x^4 Y^7}{\sqrt[3]{x^{10} Y^4}} 3 X 10 Y 4 ​ X 4 Y 7 ​ ?A. X Y 6 X Y^6 X Y 6 B. X 9 Y 9 X^9 Y^9 X 9 Y 9 C. X 8 Y 9 X 2 Y 2 3 X^8 Y^9 \sqrt[3]{x^2 Y^2} X 8 Y 9 3 X 2 Y 2 ​ D. Y 5 X 2 Y 2 3 Y^5 \sqrt[3]{x^2 Y^2} Y 5 3 X 2 Y 2 ​

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Understanding the Problem

To simplify the given expression, we need to apply the rules of exponents and radicals. The expression involves a fraction with a power of $x$ and $y$ in the numerator and a cube root in the denominator. Our goal is to rewrite the expression in its simplest form.

Simplifying the Expression

The first step is to simplify the cube root in the denominator. We can rewrite $\sqrt[3]{x^{10} y^4}$ as $x^{10/3} y^{4/3}$ using the rule $\sqrt[n]{a^n} = a$. Now, we can rewrite the original expression as:

$$\frac{x^4 y^7}{x^{10/3} y^{4/3}}$$

Applying the Quotient Rule

To simplify the expression further, we can apply the quotient rule, which states that $\frac{a^m}{a^n} = a^{m-n}$. We can rewrite the expression as:

$$x^{4 - 10/3} y^{7 - 4/3}$$

Simplifying the Exponents

To simplify the exponents, we need to find a common denominator. The least common multiple of 3 and 1 is 3. We can rewrite the exponents as:

$$x^{12/3 - 10/3} y^{21/3 - 4/3}$$

Combining the Exponents

Now, we can combine the exponents by subtracting the numerators:

$$x^{2/3} y^{17/3}$$

Rationalizing the Denominator

To rationalize the denominator, we need to multiply the expression by a clever form of 1. We can multiply the expression by $x^{2/3} y^{-2/3}$:

$$x^{2/3} y^{17/3} \cdot x^{2/3} y^{-2/3}$$

Simplifying the Expression

Now, we can simplify the expression by combining the exponents:

$$x^{2/3 + 2/3} y^{17/3 - 2/3}$$

Combining the Exponents

Now, we can combine the exponents by adding the numerators:

$$x^{4/3} y^{15/3}$$

Simplifying the Expression

To simplify the expression further, we can rewrite the exponents as:

$$x^{4/3} y^5$$

Rationalizing the Denominator

To rationalize the denominator, we need to multiply the expression by a clever form of 1. We can multiply the expression by $x^{-4/3} y^{-5}$:

$$x^{4/3} y^5 \cdot x^{-4/3} y^{-5}$$

Simplifying the Expression

Now, we can simplify the expression by combining the exponents:

$$x^{4/3 - 4/3} y^{5 - 5}$$

Combining the Exponents

Now, we can combine the exponents by subtracting the numerators:

$$x^0 y^0$$

Simplifying the Expression

To simplify the expression further, we can rewrite the exponents as:

$$1 \cdot 1$$

The Final Answer

The final answer is $\boxed{x^8 y^9 \sqrt[3]{x^2 y^2}}$.

Conclusion

In this article, we simplified the expression $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$ by applying the rules of exponents and radicals. We used the quotient rule to simplify the expression and then rationalized the denominator to get the final answer. The final answer is $\boxed{x^8 y^9 \sqrt[3]{x^2 y^2}}$.

Frequently Asked Questions

• What is the simplest form of $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$?
• How do you simplify the expression $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$?
• What is the final answer to the expression $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$?

Step-by-Step Solution

1. Simplify the cube root in the denominator.
2. Apply the quotient rule to simplify the expression.
3. Simplify the exponents by finding a common denominator.
4. Combine the exponents by subtracting the numerators.
5. Rationalize the denominator by multiplying the expression by a clever form of 1.
6. Simplify the expression by combining the exponents.
7. Rewrite the exponents as a product of powers.
8. Simplify the expression further by combining the exponents.
9. The final answer is $\boxed{x^8 y^9 \sqrt[3]{x^2 y^2}}$.

Key Concepts

• Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
• Rationalizing the denominator: multiplying the expression by a clever form of 1
• Simplifying exponents: finding a common denominator and combining the exponents

See Also

• Simplifying expressions with exponents and radicals
• Rationalizing the denominator
• Quotient rule

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
  

Frequently Asked Questions

Q: What is the simplest form of $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$?

A: The simplest form of $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$ is $x^8 y^9 \sqrt[3]{x^2 y^2}$.

Q: How do you simplify the expression $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$?

A: To simplify the expression $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$, you need to apply the quotient rule, simplify the exponents, and rationalize the denominator.

Q: What is the final answer to the expression $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$?

A: The final answer to the expression $\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}$ is $x^8 y^9 \sqrt[3]{x^2 y^2}$.

Q: How do you simplify exponents with different bases?

A: To simplify exponents with different bases, you need to find a common base and then apply the quotient rule.

Q: What is the rule for simplifying exponents with the same base?

A: The rule for simplifying exponents with the same base is to add the exponents.

Q: How do you rationalize the denominator of an expression?

A: To rationalize the denominator of an expression, you need to multiply the expression by a clever form of 1.

Q: What is the final answer to the expression $\frac{x^2 y^3}{\sqrt[3]{x^6 y^2}}$?

A: The final answer to the expression $\frac{x^2 y^3}{\sqrt[3]{x^6 y^2}}$ is $x^2 y^3 \sqrt[3]{x^2 y^2}$.

Q: How do you simplify the expression $\frac{x^3 y^4}{\sqrt[3]{x^9 y^3}}$?

A: To simplify the expression $\frac{x^3 y^4}{\sqrt[3]{x^9 y^3}}$, you need to apply the quotient rule, simplify the exponents, and rationalize the denominator.

Q: What is the final answer to the expression $\frac{x^3 y^4}{\sqrt[3]{x^9 y^3}}$?

A: The final answer to the expression $\frac{x^3 y^4}{\sqrt[3]{x^9 y^3}}$ is $x^3 y^4 \sqrt[3]{x^3 y^3}$.

Step-by-Step Solutions

Q: Simplify the expression $\frac{x^2 y^3}{\sqrt[3]{x^6 y^2}}$

A: To simplify the expression $\frac{x^2 y^3}{\sqrt[3]{x^6 y^2}}$, you need to apply the quotient rule, simplify the exponents, and rationalize the denominator.

1. Simplify the cube root in the denominator.
2. Apply the quotient rule to simplify the expression.
3. Simplify the exponents by finding a common denominator.
4. Combine the exponents by subtracting the numerators.
5. Rationalize the denominator by multiplying the expression by a clever form of 1.
6. Simplify the expression by combining the exponents.
7. The final answer is $x^2 y^3 \sqrt[3]{x^2 y^2}$.

Q: Simplify the expression $\frac{x^3 y^4}{\sqrt[3]{x^9 y^3}}$

A: To simplify the expression $\frac{x^3 y^4}{\sqrt[3]{x^9 y^3}}$, you need to apply the quotient rule, simplify the exponents, and rationalize the denominator.

1. Simplify the cube root in the denominator.
2. Apply the quotient rule to simplify the expression.
3. Simplify the exponents by finding a common denominator.
4. Combine the exponents by subtracting the numerators.
5. Rationalize the denominator by multiplying the expression by a clever form of 1.
6. Simplify the expression by combining the exponents.
7. The final answer is $x^3 y^4 \sqrt[3]{x^3 y^3}$.

Key Concepts

• Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
• Rationalizing the denominator: multiplying the expression by a clever form of 1
• Simplifying exponents: finding a common denominator and combining the exponents

See Also

• Simplifying expressions with exponents and radicals
• Rationalizing the denominator
• Quotient rule

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer