Simplify The Expression: 26 A 2 B − 5 C 9 − 4 A − 6 B C 9 \frac{26 A^2 B^{-5} C^9}{-4 A^{-6} B C^9} − 4 A − 6 B C 9 26 A 2 B − 5 C 9 A) − 13 A 8 2 B 6 \frac{-13 A^8}{2 B^6} 2 B 6 − 13 A 8 B) 13 A 8 2 B 6 \frac{13 A^8}{2 B^6} 2 B 6 13 A 8 C) 13 A 8 20 6 C \frac{13 A^8}{20^6 C} 2 0 6 C 13 A 8 D) 13 A 8 C 2 B 6 \frac{13 A^8 C}{2 B^6} 2 B 6 13 A 8 C

Mar 8, 2025 by ADMIN 358 views

Simplify the Expression: A Step-by-Step Guide

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Understanding the Problem

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and procedures involved. In this article, we will focus on simplifying the given expression: $\frac{26 a^2 b^{-5} c^9}{-4 a^{-6} b c^9}$ . We will break down the problem step by step, applying the rules of exponents and simplifying the expression to its simplest form.

The Rules of Exponents

Before we dive into the problem, let's review the rules of exponents. When we have a power raised to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ . When we have a product of powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m + n}$ . When we have a quotient of powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m - n}$ .

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the given expression. We will start by applying the rule for quotients of powers with the same base.

$\frac{26 a^2 b^{-5} c^9}{-4 a^{-6} b c^9} = \frac{26}{-4} \cdot \frac{a^2}{a^{-6}} \cdot \frac{b^{-5}}{b} \cdot \frac{c^9}{c^9}$

Applying the Rules of Exponents

Now that we have applied the rule for quotients of powers with the same base, let's simplify the expression further by applying the rules of exponents.

$\frac{26}{-4} \cdot \frac{a^2}{a^{-6}} \cdot \frac{b^{-5}}{b} \cdot \frac{c^9}{c^9} = -\frac{26}{4} \cdot a^{2 - (-6)} \cdot b^{-5 - 1} \cdot c^{9 - 9}$

Simplifying the Coefficients

Now that we have applied the rules of exponents, let's simplify the coefficients.

$-\frac{26}{4} = -\frac{13}{2}$

Simplifying the Exponents

Now that we have simplified the coefficients, let's simplify the exponents.

$a^{2 - (-6)} = a^{2 + 6} = a^8$

$b^{-5 - 1} = b^{-6}$

$c^{9 - 9} = c^0 = 1$

Combining the Results

Now that we have simplified the coefficients and the exponents, let's combine the results.

$-\frac{13}{2} \cdot a^8 \cdot b^{-6} \cdot 1 = -\frac{13 a^8}{2 b^6}$

Conclusion

In conclusion, the simplified expression is $-\frac{13 a^8}{2 b^6}$ . This is the correct answer.

Final Answer

The final answer is $\boxed{-\frac{13 a^8}{2 b^6}}$ .

Discussion

This problem requires a good understanding of the rules of exponents and how to apply them to simplify algebraic expressions. It's essential to follow the order of operations and to simplify the coefficients and the exponents separately before combining the results.

Common Mistakes

When simplifying algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

Not following the order of operations
Not simplifying the coefficients and the exponents separately
Not applying the rules of exponents correctly

Tips and Tricks

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

Make sure to follow the order of operations
Simplify the coefficients and the exponents separately
Apply the rules of exponents correctly
Use the correct notation and symbols

Practice Problems

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

Simplify the expression: $\frac{16 a^3 b^2 c^4}{-8 a^2 b c^3}$
Simplify the expression: $\frac{25 a^4 b^3 c^5}{-10 a^3 b^2 c^4}$
Simplify the expression: $\frac{36 a^5 b^4 c^6}{-18 a^4 b^3 c^5}$

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics. It's essential to understand the rules of exponents and how to apply them to simplify expressions. By following the order of operations and simplifying the coefficients and the exponents separately, you can simplify algebraic expressions with ease.

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Frequently Asked Questions

Q: What are the rules of exponents?

A: The rules of exponents are a set of rules that govern how to simplify expressions with exponents. The main rules are:

When we have a power raised to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .
When we have a product of powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m + n}$ .
When we have a quotient of powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m - n}$ .

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, follow these steps:

Identify the base and the exponent.
Apply the rules of exponents to simplify the expression.
Simplify the coefficients and the exponents separately.
Combine the results to get the final simplified expression.

Q: What is the order of operations?

A: The order of operations is a set of rules that govern how to simplify expressions with multiple operations. The main rules are:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate expressions with exponents next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Evaluate addition and subtraction operations from left to right.

Q: How do I simplify a fraction with exponents?

A: To simplify a fraction with exponents, follow these steps:

Identify the numerator and the denominator.
Apply the rules of exponents to simplify the numerator and the denominator separately.
Simplify the coefficients and the exponents in the numerator and the denominator separately.
Combine the results to get the final simplified fraction.

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

A: Some common mistakes to avoid when simplifying expressions with exponents are:

Not following the order of operations.
Not simplifying the coefficients and the exponents separately.
Not applying the rules of exponents correctly.
Not using the correct notation and symbols.

Q: How can I practice simplifying expressions with exponents?

A: You can practice simplifying expressions with exponents by:

Working through practice problems.
Using online resources and tools.
Asking a teacher or tutor for help.
Joining a study group or online community.

Additional Resources

Khan Academy: Exponents and Exponential Functions
Mathway: Exponents and Exponential Functions
Wolfram Alpha: Exponents and Exponential Functions

Conclusion

In conclusion, simplifying expressions with exponents is a crucial skill in mathematics. By understanding the rules of exponents and following the order of operations, you can simplify expressions with ease. Remember to practice regularly and seek help when needed to become proficient in simplifying expressions with exponents.

Final Tips

Make sure to follow the order of operations.
Simplify the coefficients and the exponents separately.
Apply the rules of exponents correctly.
Use the correct notation and symbols.
Practice regularly to become proficient in simplifying expressions with exponents.

Commonly Asked Questions

Q: What is the difference between a power and an exponent?

A: A power is the result of raising a number to a certain power, while an exponent is the number that is raised to a certain power.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, follow the same steps as simplifying an expression with a single exponent, but apply the rules of exponents multiple times.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is raised to a certain power, while a negative exponent indicates that the base is raised to a certain power and then taken as a reciprocal.

Q: How do I simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, the result is always 1, regardless of the base.