Simplify The Expression:$\[ \frac{a P Q \times \left(a P^2 Q^2\right)^2}{32 P^6 P^3} \\]

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Introduction

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In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. One such expression that requires simplification is $\frac{a p q \times \left(a p^2 q^2\right)^2}{32 p^6 p^3}$. In this article, we will break down this expression and simplify it step by step.

Understanding the Expression

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Before we dive into simplifying the expression, let's understand what it represents. The expression is a fraction with two parts: the numerator and the denominator. The numerator is the product of $a$, $p$, $q$, and the square of $\left(a p^2 q^2\right)$. The denominator is $32$ multiplied by $p^6$ and $p^3$.

Simplifying the Numerator

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The numerator can be simplified by first expanding the square of $\left(a p^2 q^2\right)$. Using the exponent rule $\left(ab\right)^n = a^n b^n$, we can rewrite the numerator as:

$$a p q \times \left(a p^2 q^2\right)^2 = a p q \times a^2 p^4 q^4$$

Applying Exponent Rules

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Now, let's apply the exponent rule $a^m \times a^n = a^{m+n}$ to simplify the numerator further:

$$a p q \times a^2 p^4 q^4 = a^{1+2} p^{1+4} q^{1+4} = a^3 p^5 q^5$$

Simplifying the Denominator

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The denominator can be simplified by first combining the powers of $p$ using the exponent rule $a^m \times a^n = a^{m+n}$:

$$32 p^6 p^3 = 32 p^{6+3} = 32 p^9$$

Putting it All Together

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Now that we have simplified the numerator and the denominator, we can put them together to get the final simplified expression:

$$\frac{a^3 p^5 q^5}{32 p^9}$$

Canceling Out Common Factors

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Before we can simplify the expression further, we need to cancel out any common factors between the numerator and the denominator. In this case, we can cancel out $p^5$ from both the numerator and the denominator:

$$\frac{a^3 q^5}{32 p^4}$$

Final Simplification

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The expression is now simplified, but we can take it a step further by simplifying the fraction. We can divide both the numerator and the denominator by their greatest common divisor, which is $1$ in this case. However, we can simplify the fraction by dividing both the numerator and the denominator by $p^4$:

$$\frac{a^3 q^5}{32 p^4} = \frac{a^3 q^5}{32} \times \frac{1}{p^4}$$

Conclusion

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In conclusion, simplifying the expression $\frac{a p q \times \left(a p^2 q^2\right)^2}{32 p^6 p^3}$ requires a step-by-step approach. By understanding the expression, simplifying the numerator and the denominator, canceling out common factors, and simplifying the fraction, we can arrive at the final simplified expression: $\frac{a^3 q^5}{32 p^4}$. This expression is a crucial concept in mathematics, and understanding it can help us solve complex problems and develop a deeper appreciation for the subject.

Frequently Asked Questions

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Q: What is the simplified expression?

A: The simplified expression is $\frac{a^3 q^5}{32 p^4}$.

Q: How do I simplify the expression?

A: To simplify the expression, you need to follow these steps: understand the expression, simplify the numerator and the denominator, cancel out common factors, and simplify the fraction.

Q: What is the greatest common divisor of the numerator and the denominator?

A: The greatest common divisor of the numerator and the denominator is $1$.

Q: Can I simplify the fraction further?

A: Yes, you can simplify the fraction further by dividing both the numerator and the denominator by their greatest common divisor, which is $1$ in this case.

Final Thoughts

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Simplifying expressions is a crucial skill in mathematics that helps us solve complex problems and understand the underlying concepts. By following the steps outlined in this article, you can simplify the expression $\frac{a p q \times \left(a p^2 q^2\right)^2}{32 p^6 p^3}$ and arrive at the final simplified expression: $\frac{a^3 q^5}{32 p^4}$. This expression is a fundamental concept in mathematics, and understanding it can help you develop a deeper appreciation for the subject.

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Q&A: Simplifying the Expression

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Q: What is the simplified expression?

A: The simplified expression is $\frac{a^3 q^5}{32 p^4}$.

Q: How do I simplify the expression?

A: To simplify the expression, you need to follow these steps:

1. Understand the expression: Before you can simplify the expression, you need to understand what it represents.
2. Simplify the numerator: The numerator can be simplified by expanding the square of $\left(a p^2 q^2\right)$ and applying exponent rules.
3. Simplify the denominator: The denominator can be simplified by combining the powers of $p$ using exponent rules.
4. Cancel out common factors: Once you have simplified the numerator and the denominator, you can cancel out any common factors between the two.
5. Simplify the fraction: Finally, you can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor.

Q: What is the greatest common divisor of the numerator and the denominator?

A: The greatest common divisor of the numerator and the denominator is $1$.

Q: Can I simplify the fraction further?

A: Yes, you can simplify the fraction further by dividing both the numerator and the denominator by their greatest common divisor, which is $1$ in this case.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not understanding the expression: Before you can simplify the expression, you need to understand what it represents.
• Not simplifying the numerator and denominator separately: Simplifying the numerator and denominator separately can help you avoid mistakes and ensure that you are simplifying the expression correctly.
• Not canceling out common factors: Failing to cancel out common factors between the numerator and the denominator can result in an incorrect simplified expression.

Q: How do I know if I have simplified the expression correctly?

A: To ensure that you have simplified the expression correctly, you can:

• Check your work: Double-check your work to ensure that you have simplified the expression correctly.
• Use a calculator: If you are unsure about the simplified expression, you can use a calculator to check your work.
• Ask for help: If you are still unsure about the simplified expression, you can ask for help from a teacher or tutor.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

• Science and engineering: Simplifying expressions is used in science and engineering to solve complex problems and understand the underlying concepts.
• Finance: Simplifying expressions is used in finance to calculate interest rates and investment returns.
• Computer programming: Simplifying expressions is used in computer programming to write efficient and effective code.

Conclusion

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Simplifying expressions is a crucial skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can simplify the expression $\frac{a p q \times \left(a p^2 q^2\right)^2}{32 p^6 p^3}$ and arrive at the final simplified expression: $\frac{a^3 q^5}{32 p^4}$. Remember to understand the expression, simplify the numerator and the denominator, cancel out common factors, and simplify the fraction to ensure that you are simplifying the expression correctly.

Frequently Asked Questions

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Q: What is the simplified expression?

A: The simplified expression is $\frac{a^3 q^5}{32 p^4}$.

Q: How do I simplify the expression?

A: To simplify the expression, you need to follow these steps: understand the expression, simplify the numerator and the denominator, cancel out common factors, and simplify the fraction.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include not understanding the expression, not simplifying the numerator and denominator separately, and not canceling out common factors.

Q: How do I know if I have simplified the expression correctly?

A: To ensure that you have simplified the expression correctly, you can check your work, use a calculator, and ask for help.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including science and engineering, finance, and computer programming.

Final Thoughts

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Simplifying expressions is a crucial skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can simplify the expression $\frac{a p q \times \left(a p^2 q^2\right)^2}{32 p^6 p^3}$ and arrive at the final simplified expression: $\frac{a^3 q^5}{32 p^4}$. Remember to understand the expression, simplify the numerator and the denominator, cancel out common factors, and simplify the fraction to ensure that you are simplifying the expression correctly.