Not factoring out the coefficients

Oct 13, 2025 by miodeya.com 117 views

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given problem: $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ . We will break down the solution into manageable steps, using the properties of exponents and fractions to arrive at the final answer.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $a^3$ can be read as "a to the power of 3" and is equivalent to $a \times a \times a$ . When simplifying algebraic expressions, it's essential to understand the rules of exponents, including:

Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, $a^2 \times a^3 = a^{2+3} = a^5$ .
Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(a^2)^3 = a^{2 \times 3} = a^6$ .
Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. For example, $\frac{a^2}{a^3} = a^{2-3} = a^{-1}$ .

Simplifying the Given Expression

Now that we have a solid understanding of exponents, let's apply these rules to simplify the given expression: $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ .

Step 1: Factor Out the Coefficients

The first step in simplifying the expression is to factor out the coefficients. In this case, we can factor out 28 and 12:

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

Step 2: Simplify the Coefficients

Next, we simplify the coefficients by dividing 28 by 12:

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

Step 3: Apply the Quotient of Powers Rule

Now, we apply the quotient of powers rule to simplify the exponents:

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

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

Step 4: Combine the Simplified Terms

Finally, we combine the simplified terms:

\frac{7}{3} \times p^{15} \times q^{-12} = \frac{7 p^{15} q^{-12}}{3}

Conclusion

In conclusion, the simplified expression is $\frac{7 p^{15} q^{-12}}{3}$ . This can be rewritten as $\frac{7 p^{15}}{3 q^{12}}$ , which is the correct answer.

Final Answer

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

Discussion

This problem requires a deep understanding of exponents and fractions. The key to simplifying the expression is to apply the quotient of powers rule and to factor out the coefficients. By following these steps, we can arrive at the final answer.

Common Mistakes

When simplifying algebraic expressions, it's essential to avoid common mistakes, such as:

Not factoring out the coefficients: Failing to factor out the coefficients can lead to incorrect simplification.
Not applying the quotient of powers rule: Failing to apply the quotient of powers rule can result in incorrect exponents.
Not combining the simplified terms: Failing to combine the simplified terms can lead to an incorrect final answer.

Tips and Tricks

When simplifying algebraic expressions, here are some tips and tricks to keep in mind:

Use the product of powers rule: When multiplying two powers with the same base, add the exponents.
Use the power of a power rule: When raising a power to another power, multiply the exponents.
Use the quotient of powers rule: When dividing two powers with the same base, subtract the exponents.
Factor out the coefficients: Factor out the coefficients to simplify the expression.
Combine the simplified terms: Combine the simplified terms to arrive at the final answer.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

$\frac{24 x^5 y^{-3}}{8 x^{-2} y^4}$
$\frac{36 z^7 w^{-2}}{12 z^{-3} w^5}$
$\frac{48 r^9 s^{-6}}{16 r^{-4} s^3}$

By following the steps outlined in this article and practicing with these problems, you'll become proficient in simplifying algebraic expressions in no time!

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given problem: $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ . We broke down the solution into manageable steps, using the properties of exponents and fractions to arrive at the final answer. In this article, we'll answer some frequently asked questions about simplifying algebraic expressions.

Q&A

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

A: The first step in simplifying an algebraic expression is to factor out the coefficients. This involves separating the numerical coefficients from the variables.

Q: How do I simplify exponents when multiplying two powers with the same base?

A: When multiplying two powers with the same base, add the exponents. For example, $a^2 \times a^3 = a^{2+3} = a^5$ .

Q: How do I simplify exponents when raising a power to another power?

A: When raising a power to another power, multiply the exponents. For example, $(a^2)^3 = a^{2 \times 3} = a^6$ .

Q: How do I simplify exponents when dividing two powers with the same base?

A: When dividing two powers with the same base, subtract the exponents. For example, $\frac{a^2}{a^3} = a^{2-3} = a^{-1}$ .

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two powers with the same base, subtract the exponents. For example, $\frac{a^2}{a^3} = a^{2-3} = a^{-1}$ .

Q: How do I combine the simplified terms?

A: To combine the simplified terms, multiply the coefficients and add the exponents. For example, $\frac{7 p^{15} q^{-12}}{3} = \frac{7 p^{15}}{3 q^{12}}$ .

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

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

Not factoring out the coefficients
Not applying the quotient of powers rule
Not combining the simplified terms

Q: What are some tips and tricks for simplifying algebraic expressions?

A: Some tips and tricks for simplifying algebraic expressions include:

Using the product of powers rule
Using the power of a power rule
Using the quotient of powers rule
Factoring out the coefficients
Combining the simplified terms