Find ( 3 X 2 Y − 5 ) 3 \left(3 X^2 Y^{-5}\right)^3 ( 3 X 2 Y − 5 ) 3 .A) 27 X 6 Y 15 27 X^6 Y^{15} 27 X 6 Y 15 B) 27 X 6 Y − 15 27 X^6 Y^{-15} 27 X 6 Y − 15 C) 9 X 6 Y − 15 9 X^6 Y^{-15} 9 X 6 Y − 15 D) 27 X 5 Y 2 27 X^5 Y^2 27 X 5 Y 2

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Understanding the Problem

To find the value of $\left(3 x^2 y^{-5}\right)^3$, we need to apply the rules of exponents. The expression inside the parentheses is raised to the power of 3, which means we need to multiply the exponent of each variable by 3.

Applying the Rules of Exponents

When a power is raised to another power, we multiply the exponents. In this case, we have:

$\left(3 x^2 y^{-5}\right)^3 = 3^3 \cdot \left(x^2\right)^3 \cdot \left(y^{-5}\right)^3$

Simplifying the Expression

Now, we can simplify the expression by applying the rules of exponents:

$3^3 = 27$

$\left(x^2\right)^3 = x^{2 \cdot 3} = x^6$

$\left(y^{-5}\right)^3 = y^{-5 \cdot 3} = y^{-15}$

Combining the Terms

Now, we can combine the terms to get the final answer:

$\left(3 x^2 y^{-5}\right)^3 = 27 x^6 y^{-15}$

Conclusion

Therefore, the correct answer is $\boxed{B) 27 x^6 y^{-15}}$.

Discussion

This problem requires a good understanding of the rules of exponents. The key concept is that when a power is raised to another power, we multiply the exponents. This is a fundamental concept in algebra and is used extensively in mathematics.

Common Mistakes

One common mistake is to forget to multiply the exponents when a power is raised to another power. This can lead to incorrect answers. Another common mistake is to get the sign of the exponent wrong. In this case, the exponent of y is negative, so we need to make sure to get the sign correct.

Real-World Applications

This concept is used extensively in mathematics and has many real-world applications. For example, in physics, the laws of motion and energy can be expressed using exponents. In economics, the concept of compound interest can be expressed using exponents.

Practice Problems

Here are a few practice problems to help you understand the concept better:

1. Find $\left(2 x^3 y^2\right)^4$
2. Find $\left(3 x^2 y^{-3}\right)^2$
3. Find $\left(4 x^4 y^3\right)^3$

Solutions

1. $\left(2 x^3 y^2\right)^4 = 2^4 \cdot \left(x^3\right)^4 \cdot \left(y^2\right)^4 = 16 x^{12} y^8$
2. $\left(3 x^2 y^{-3}\right)^2 = 3^2 \cdot \left(x^2\right)^2 \cdot \left(y^{-3}\right)^2 = 9 x^4 y^{-6}$
3. $\left(4 x^4 y^3\right)^3 = 4^3 \cdot \left(x^4\right)^3 \cdot \left(y^3\right)^3 = 64 x^{12} y^9$

Conclusion

In conclusion, finding the value of $\left(3 x^2 y^{-5}\right)^3$ requires a good understanding of the rules of exponents. The key concept is that when a power is raised to another power, we multiply the exponents. This concept is used extensively in mathematics and has many real-world applications.

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Frequently Asked Questions

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

A: When a power is raised to another power, we multiply the exponents. This means that if we have an expression of the form $(a^m)^n$, we can simplify it to $a^{m \cdot n}$.

Q: How do we apply the rule to the given expression?

A: To find the value of $\left(3 x^2 y^{-5}\right)^3$, we need to apply the rule by multiplying the exponents. This gives us:

$\left(3 x^2 y^{-5}\right)^3 = 3^3 \cdot \left(x^2\right)^3 \cdot \left(y^{-5}\right)^3$

Q: What is the value of $3^3$?

A: $3^3 = 27$.

Q: What is the value of $\left(x^2\right)^3$?

A: $\left(x^2\right)^3 = x^{2 \cdot 3} = x^6$.

Q: What is the value of $\left(y^{-5}\right)^3$?

A: $\left(y^{-5}\right)^3 = y^{-5 \cdot 3} = y^{-15}$.

Q: How do we combine the terms to get the final answer?

A: Now that we have the values of $3^3$, $\left(x^2\right)^3$, and $\left(y^{-5}\right)^3$, we can combine them to get the final answer:

$\left(3 x^2 y^{-5}\right)^3 = 27 x^6 y^{-15}$.

Q: What is the correct answer?

A: The correct answer is $\boxed{B) 27 x^6 y^{-15}}$.

Q: What are some common mistakes to avoid?

A: One common mistake is to forget to multiply the exponents when a power is raised to another power. Another common mistake is to get the sign of the exponent wrong. In this case, the exponent of y is negative, so we need to make sure to get the sign correct.

Q: What are some real-world applications of this concept?

A: This concept is used extensively in mathematics and has many real-world applications. For example, in physics, the laws of motion and energy can be expressed using exponents. In economics, the concept of compound interest can be expressed using exponents.

Q: How can I practice this concept?

A: Here are a few practice problems to help you understand the concept better:

1. Find $\left(2 x^3 y^2\right)^4$
2. Find $\left(3 x^2 y^{-3}\right)^2$
3. Find $\left(4 x^4 y^3\right)^3$

Q: What are the solutions to the practice problems?

A: Here are the solutions to the practice problems:

1. $\left(2 x^3 y^2\right)^4 = 2^4 \cdot \left(x^3\right)^4 \cdot \left(y^2\right)^4 = 16 x^{12} y^8$
2. $\left(3 x^2 y^{-3}\right)^2 = 3^2 \cdot \left(x^2\right)^2 \cdot \left(y^{-3}\right)^2 = 9 x^4 y^{-6}$
3. $\left(4 x^4 y^3\right)^3 = 4^3 \cdot \left(x^4\right)^3 \cdot \left(y^3\right)^3 = 64 x^{12} y^9$

Conclusion

In conclusion, finding the value of $\left(3 x^2 y^{-5}\right)^3$ requires a good understanding of the rules of exponents. The key concept is that when a power is raised to another power, we multiply the exponents. This concept is used extensively in mathematics and has many real-world applications.