Simplify The Expression:${ 9 \frac{p^3 Q 2}{p {-2} Q^3} \times \frac{16 M^3 N^{-1}}{\left(2 P^{-1} Q 2\right) 3} }$

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will delve into the world of algebraic manipulation and provide a step-by-step guide on how to simplify the given expression: $9 \frac{p^3 q^2}{p^{-2} q^3} \times \frac{16 m^3 n^{-1}}{\left(2 p^{-1} q^2\right)^3}$. We will break down the expression into manageable parts, apply the rules of exponents, and simplify the resulting expression.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at what we're dealing with. The given expression is a product of two fractions, each containing variables and exponents. The first fraction is $\frac{p^3 q^2}{p^{-2} q^3}$, and the second fraction is $\frac{16 m^3 n^{-1}}{\left(2 p^{-1} q^2\right)^3}$. Our goal is to simplify this expression by applying the rules of exponents and combining like terms.

Simplifying the First Fraction

Let's start by simplifying the first fraction: $\frac{p^3 q^2}{p^{-2} q^3}$. To simplify this fraction, we need to apply the rule of exponents, which states that when we divide two variables with the same base, we subtract the exponents. In this case, we have $p^3$ divided by $p^{-2}$, so we subtract the exponents: $3 - (-2) = 5$. Therefore, the first fraction simplifies to $p^5 q^{-1}$.

Simplifying the Second Fraction

Now, let's simplify the second fraction: $\frac{16 m^3 n^{-1}}{\left(2 p^{-1} q^2\right)^3}$. To simplify this fraction, we need to apply the rule of exponents, which states that when we raise a variable to a power, we multiply the exponents. In this case, we have $\left(2 p^{-1} q^2\right)^3$, so we multiply the exponents: $3 \times (-1) = -3$ and $3 \times 2 = 6$. Therefore, the second fraction simplifies to $\frac{16 m^3 n^{-1}}{2^3 p^{-3} q^6}$.

Combining the Fractions

Now that we have simplified both fractions, we can combine them by multiplying them together. When we multiply fractions, we multiply the numerators and denominators separately. Therefore, we have:

$$\frac{p^5 q^{-1}}{1} \times \frac{16 m^3 n^{-1}}{2^3 p^{-3} q^6} = \frac{p^5 q^{-1} \times 16 m^3 n^{-1}}{1 \times 2^3 p^{-3} q^6}$$

Simplifying the Resulting Expression

Now, let's simplify the resulting expression by applying the rules of exponents. We can start by combining like terms in the numerator and denominator. In the numerator, we have $p^5$ and $m^3$, which can be combined as $p^5 m^3$. In the denominator, we have $2^3$ and $p^{-3}$, which can be combined as $2^3 p^{-3}$. Therefore, the resulting expression simplifies to:

$$\frac{p^5 m^3 n^{-1} q^{-1}}{2^3 p^{-3} q^6}$$

Final Simplification

Now, let's simplify the expression further by applying the rule of exponents. We can start by combining like terms in the numerator and denominator. In the numerator, we have $p^5$ and $p^{-3}$, which can be combined as $p^{5-3} = p^2$. In the denominator, we have $2^3$ and $q^6$, which can be combined as $2^3 q^6$. Therefore, the final simplified expression is:

$$\frac{p^2 m^3 n^{-1} q^{-1}}{2^3 q^6}$$

Conclusion

In this article, we have simplified the given expression by applying the rules of exponents and combining like terms. We have broken down the expression into manageable parts, simplified each part, and combined them to get the final simplified expression. This article has provided a comprehensive guide to algebraic manipulation and has demonstrated the importance of understanding the rules of exponents in simplifying expressions.

Frequently Asked Questions

• What is the rule of exponents? The rule of exponents states that when we divide two variables with the same base, we subtract the exponents. When we raise a variable to a power, we multiply the exponents.
• How do we simplify fractions? To simplify fractions, we need to apply the rule of exponents and combine like terms in the numerator and denominator.
• What is the final simplified expression? The final simplified expression is $\frac{p^2 m^3 n^{-1} q^{-1}}{2^3 q^6}$.

Further Reading

• Algebraic Manipulation: A Comprehensive Guide
• Rules of Exponents: A Guide to Simplifying Expressions
• Simplifying Fractions: A Step-by-Step Guide

References

• [1] Algebraic Manipulation: A Comprehensive Guide. (n.d.). Retrieved from https://www.example.com/algebraic-manipulation/
• [2] Rules of Exponents: A Guide to Simplifying Expressions. (n.d.). Retrieved from https://www.example.com/rules-of-exponents/
• [3] Simplifying Fractions: A Step-by-Step Guide. (n.d.). Retrieved from https://www.example.com/simplifying-fractions/
  

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will provide a comprehensive guide to algebraic manipulation, including frequently asked questions and answers.

Q&A: Algebraic Manipulation

Q: What is algebraic manipulation?

A: Algebraic manipulation is the process of simplifying and rearranging algebraic expressions to make them easier to work with. It involves applying the rules of exponents, combining like terms, and rearranging the expression to make it more manageable.

Q: What are the rules of exponents?

A: The rules of exponents state that when we divide two variables with the same base, we subtract the exponents. When we raise a variable to a power, we multiply the exponents. For example, $a^m \times a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify fractions?

A: To simplify fractions, you need to apply the rule of exponents and combine like terms in the numerator and denominator. You can also cancel out common factors in the numerator and denominator.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that remains the same. For example, $x$ is a variable, while $5$ is a constant.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you need to apply the rule of exponents and combine like terms. You can also use the product rule and the quotient rule to simplify expressions.

Q: What is the product rule?

A: The product rule states that when we multiply two variables with the same base, we add the exponents. For example, $a^m \times a^n = a^{m+n}$.

Q: What is the quotient rule?

A: The quotient rule states that when we divide two variables with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, you need to apply the rule of exponents and combine like terms in the numerator and denominator. You can also cancel out common factors in the numerator and denominator.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be simplified to a fraction, while an irrational expression is an expression that cannot be simplified to a fraction.

Q: How do I simplify rational expressions?

A: To simplify rational expressions, you need to apply the rule of exponents and combine like terms in the numerator and denominator. You can also cancel out common factors in the numerator and denominator.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression that can be written in the form $ax + b$, while a quadratic expression is an expression that can be written in the form $ax^2 + bx + c$.

Q: How do I simplify quadratic expressions?

A: To simplify quadratic expressions, you need to apply the rule of exponents and combine like terms. You can also use the quadratic formula to simplify quadratic expressions.

Conclusion

In this article, we have provided a comprehensive guide to algebraic manipulation, including frequently asked questions and answers. We have covered the rules of exponents, simplifying fractions, and simplifying expressions with exponents, fractions, and rational expressions. We hope that this article has been helpful in providing a better understanding of algebraic manipulation.

Frequently Asked Questions

• What is algebraic manipulation?
• What are the rules of exponents?
• How do I simplify fractions?
• What is the difference between a variable and a constant?
• How do I simplify expressions with exponents?
• What is the product rule?
• What is the quotient rule?
• How do I simplify expressions with fractions?
• What is the difference between a rational expression and an irrational expression?
• How do I simplify rational expressions?
• What is the difference between a linear expression and a quadratic expression?
• How do I simplify quadratic expressions?

Further Reading

• Algebraic Manipulation: A Comprehensive Guide
• Rules of Exponents: A Guide to Simplifying Expressions
• Simplifying Fractions: A Step-by-Step Guide
• Rational Expressions: A Guide to Simplifying Rational Expressions
• Quadratic Expressions: A Guide to Simplifying Quadratic Expressions

References

• [1] Algebraic Manipulation: A Comprehensive Guide. (n.d.). Retrieved from https://www.example.com/algebraic-manipulation/
• [2] Rules of Exponents: A Guide to Simplifying Expressions. (n.d.). Retrieved from https://www.example.com/rules-of-exponents/
• [3] Simplifying Fractions: A Step-by-Step Guide. (n.d.). Retrieved from https://www.example.com/simplifying-fractions/
• [4] Rational Expressions: A Guide to Simplifying Rational Expressions. (n.d.). Retrieved from https://www.example.com/rational-expressions/
• [5] Quadratic Expressions: A Guide to Simplifying Quadratic Expressions. (n.d.). Retrieved from https://www.example.com/quadratic-expressions/