Which Expression Is Equivalent To 28 P 9 Q − 5 12 P − 6 Q 7 \frac{28 P^9 Q^{-5}}{12 P^{-6} Q^7} 12 P − 6 Q 7 28 P 9 Q − 5 ? Assume P ≠ 0 P \neq 0 P  = 0 And Q ≠ 0 Q \neq 0 Q  = 0 .A. 2 P 15 Q 12 \frac{2}{p^{15} Q^{12}} P 15 Q 12 2 B. 7 P 15 3 Q 12 \frac{7 P^{15}}{3 Q^{12}} 3 Q 12 7 P 15 C. 2 Q 12 P 15 \frac{2 Q^{12}}{p^{15}} P 15 2 Q 12

Mar 12, 2025 by ADMIN 349 views

Simplifying Algebraic Expressions: A Step-by-Step Guide

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ . We will assume that $p \neq 0$ and $q \neq 0$ , and we will use this assumption to simplify the expression.

Understanding the Given Expression

The given expression is $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ . This expression consists of two parts: the numerator and the denominator. The numerator is $28 p^9 q^{-5}$ , and the denominator is $12 p^{-6} q^7$ .

Breaking Down the Numerator

The numerator is $28 p^9 q^{-5}$ . This can be broken down into three parts: the coefficient $28$ , the variable part $p^9$ , and the variable part $q^{-5}$ .

Breaking Down the Denominator

The denominator is $12 p^{-6} q^7$ . This can be broken down into three parts: the coefficient $12$ , the variable part $p^{-6}$ , and the variable part $q^7$ .

Simplifying the Expression

To simplify the expression, we need to use the rules of exponents. Specifically, we need to use the rule that states that when we divide two powers with the same base, we subtract the exponents.

Using the Rule of Exponents

Using the rule of exponents, we can simplify the expression as follows:

\frac{28 p^9 q^{-5}}{12 p^{-6} q^7} = \frac{28}{12} \cdot \frac{p^9}{p^{-6}} \cdot \frac{q^{-5}}{q^7}

Simplifying the Coefficients

The coefficients are $\frac{28}{12}$ and $\frac{p^9}{p^{-6}}$ and $\frac{q^{-5}}{q^7}$ . We can simplify these coefficients as follows:

\frac{28}{12} = \frac{7}{3}

\frac{p^9}{p^{-6}} = p^{9-(-6)} = p^{15}

\frac{q^{-5}}{q^7} = q^{-5-7} = q^{-12}

Combining the Simplified Coefficients

We can now combine the simplified coefficients to get the final simplified expression:

\frac{28 p^9 q^{-5}}{12 p^{-6} q^7} = \frac{7}{3} \cdot p^{15} \cdot q^{-12}

Conclusion

In conclusion, the given expression $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ can be simplified to $\frac{7}{3} \cdot p^{15} \cdot q^{-12}$ . This expression is equivalent to $\frac{7 p^{15}}{3 q^{12}}$ .

Final Answer

The final answer is $\boxed{\frac{7 p^{15}}{3 q^{12}}}$ .

Discussion

The given expression $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ can be simplified to $\frac{7}{3} \cdot p^{15} \cdot q^{-12}$ . This expression is equivalent to $\frac{7 p^{15}}{3 q^{12}}$ . The simplification process involves using the rules of exponents and simplifying the coefficients.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Break down the numerator and denominator into their respective parts.
Use the rule of exponents to simplify the expression.
Simplify the coefficients using the rules of exponents.
Combine the simplified coefficients to get the final simplified expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

Not using the rules of exponents correctly.
Not simplifying the coefficients correctly.
Not combining the simplified coefficients correctly.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

Use the rules of exponents correctly.
Simplify the coefficients correctly.
Combine the simplified coefficients correctly.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. For example, it can be used to solve problems in physics, engineering, and economics.

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

Simplify the expression $\frac{24 x^7 y^{-3}}{8 x^{-4} y^5}$ .
Simplify the expression $\frac{36 z^9 w^{-2}}{12 z^{-6} w^3}$ .

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By using the rules of exponents and simplifying the coefficients, we can simplify complex expressions and arrive at the final answer.

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Q&A: Simplifying Algebraic Expressions

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to break down the numerator and denominator into their respective parts.

Q: What is the rule of exponents that we use to simplify expressions?

A: The rule of exponents that we use to simplify expressions is that when we divide two powers with the same base, we subtract the exponents.

Q: How do we simplify the coefficients in an algebraic expression?

A: We simplify the coefficients by using the rules of exponents. Specifically, we can simplify the coefficients by adding or subtracting the exponents.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to combine the simplified coefficients to get the final simplified expression.

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

A: Some common mistakes to avoid when simplifying algebraic expressions include not using the rules of exponents correctly, not simplifying the coefficients correctly, and not combining the simplified coefficients correctly.

Q: How can we use simplifying algebraic expressions in real-world applications?

A: Simplifying algebraic expressions has many real-world applications, including solving problems in physics, engineering, and economics.

Q: What are some practice problems that can help us practice simplifying algebraic expressions?

A: Some practice problems that can help us practice simplifying algebraic expressions include simplifying the expression $\frac{24 x^7 y^{-3}}{8 x^{-4} y^5}$ and simplifying the expression $\frac{36 z^9 w^{-2}}{12 z^{-6} w^3}$ .

Additional Tips and Tricks

Q: What is the best way to approach simplifying algebraic expressions?

A: The best way to approach simplifying algebraic expressions is to break down the numerator and denominator into their respective parts, use the rules of exponents to simplify the expression, simplify the coefficients, and combine the simplified coefficients to get the final simplified expression.

Q: How can we use technology to help us simplify algebraic expressions?

A: We can use technology, such as calculators or computer software, to help us simplify algebraic expressions. However, it's also important to understand the underlying math concepts and to be able to simplify expressions by hand.

Q: What are some common algebraic expressions that we can simplify?

A: Some common algebraic expressions that we can simplify include expressions with exponents, expressions with fractions, and expressions with variables.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article and practicing with sample problems, we can become proficient in simplifying algebraic expressions and apply this skill to real-world problems.

Final Thoughts

Simplifying algebraic expressions is a fundamental concept in mathematics, and it has many real-world applications. By understanding the rules of exponents and how to simplify coefficients, we can simplify complex expressions and arrive at the final answer. With practice and patience, we can become proficient in simplifying algebraic expressions and apply this skill to a wide range of problems.

Additional Resources

For additional resources on simplifying algebraic expressions, including practice problems and video tutorials, please visit the following websites:

Khan Academy: Algebra
Mathway: Algebra
Wolfram Alpha: Algebra