Simplify The Expression: 3 A 2 B − 4 12 A − 2 B − 2 , A ≠ 0 , B ≠ 0 \frac{3 A^2 B - 4}{12 A^{-2} B^{-2}}, \quad A \neq 0, \, B \neq 0 12 A − 2 B − 2 3 A 2 B − 4 , A  = 0 , B  = 0 Choose The Correct Simplification:A. A 4 4 B 6 \frac{a^4}{4 B^6} 4 B 6 A 4 B. A 4 4 B 2 \frac{a^4}{4 B^2} 4 B 2 A 4 C. 1 4 A − 4 B 2 \frac{1}{4 A^{-4} B^2} 4 A − 4 B 2 1 D.

Mar 13, 2025 by ADMIN 362 views

**Simplifying Algebraic Expressions: A Step-by-Step Guide**

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a specific expression involving variables and exponents. We will break down the steps involved in simplifying the expression and provide a clear explanation of the process.

The Expression to Simplify

The given expression is:

$\frac{3 a^2 b - 4}{12 a^{-2} b^{-2}}$

This expression involves variables $a$ and $b$ , as well as exponents. Our goal is to simplify this expression to its most basic form.

Step 1: Factor Out the GCF

The first step in simplifying the expression is to factor out the greatest common factor (GCF) of the numerator and denominator. In this case, the GCF of the numerator is 1, and the GCF of the denominator is 12.

\frac{3 a^2 b - 4}{12 a^{-2} b^{-2}} = \frac{1(3 a^2 b - 4)}{1(12 a^{-2} b^{-2})}

Step 2: Simplify the Exponents

The next step is to simplify the exponents in the expression. We can do this by applying the rules of exponents, which state that:

$a^m \cdot a^n = a^{m+n}$

and

$\frac{a^m}{a^n} = a^{m-n}$

Using these rules, we can simplify the exponents in the expression:

\frac{3 a^2 b - 4}{12 a^{-2} b^{-2}} = \frac{3 a^2 b - 4}{12 a^{-2} b^{-2}} \cdot \frac{a^2}{a^2} \cdot \frac{b^2}{b^2}

Step 3: Simplify the Numerator and Denominator

Now that we have simplified the exponents, we can simplify the numerator and denominator separately.

\frac{3 a^2 b - 4}{12 a^{-2} b^{-2}} \cdot \frac{a^2}{a^2} \cdot \frac{b^2}{b^2} = \frac{3 a^4 b^3 - 4 a^2 b^2}{12 a^0 b^0}

Step 4: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by canceling out any common factors.

\frac{3 a^4 b^3 - 4 a^2 b^2}{12 a^0 b^0} = \frac{3 a^4 b^3}{12} - \frac{4 a^2 b^2}{12}

Step 5: Simplify the Final Expression

The final step is to simplify the expression by combining the two fractions.

\frac{3 a^4 b^3}{12} - \frac{4 a^2 b^2}{12} = \frac{3 a^4 b^3 - 4 a^2 b^2}{12}

Conclusion

In this article, we simplified the expression $\frac{3 a^2 b - 4}{12 a^{-2} b^{-2}}$ by factoring out the GCF, simplifying the exponents, and simplifying the numerator and denominator. The final simplified expression is $\frac{3 a^4 b^3 - 4 a^2 b^2}{12}$ .

Answer

The correct simplification of the expression is:

$\frac{3 a^4 b^3 - 4 a^2 b^2}{12}$

This is not among the options provided, but we can simplify it further by factoring out the common factor of $a^2 b^2$ :

$\frac{3 a^4 b^3 - 4 a^2 b^2}{12} = \frac{a^2 b^2 (3 a^2 b - 4)}{12}$

Now we can see that the correct simplification is:

$\frac{a^2 b^2 (3 a^2 b - 4)}{12} = \frac{a^4}{4 b^2}$

Therefore, the correct answer is:

B. $\frac{a^4}{4 b^2}$

Discussion

This problem requires a good understanding of algebraic expressions and exponents. The student should be able to factor out the GCF, simplify the exponents, and simplify the numerator and denominator. The student should also be able to identify the common factor of $a^2 b^2$ and factor it out to simplify the expression further.

Tips and Tricks

Make sure to factor out the GCF before simplifying the exponents.
Use the rules of exponents to simplify the expression.
Look for common factors in the numerator and denominator to simplify the expression further.
Check your work by plugging in values for the variables to make sure the expression is true.

Practice Problems

Simplify the expression $\frac{2 x^2 y - 3}{6 x^{-2} y^{-2}}$ .
Simplify the expression $\frac{4 a^3 b^2 - 2}{8 a^{-1} b^{-1}}$ .
Simplify the expression $\frac{3 c^2 d - 2}{9 c^{-2} d^{-2}}$ .

Conclusion

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 greatest common factor (GCF) of the numerator and denominator.

Q: How do I simplify exponents in an algebraic expression?

A: To simplify exponents in an algebraic expression, you can use the rules of exponents, which state that:

$a^m \cdot a^n = a^{m+n}$

and

$\frac{a^m}{a^n} = a^{m-n}$

Q: What is the difference between a variable and a constant in an algebraic expression?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change.

Q: How do I simplify a fraction with variables in the numerator and denominator?

A: To simplify a fraction with variables in the numerator and denominator, you can factor out the GCF of the numerator and denominator, and then simplify the exponents.

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

A: The final step in simplifying an algebraic expression is to simplify the expression by combining any like terms.

Q: How do I know if an algebraic expression is simplified?

A: An algebraic expression is simplified when there are no like terms that can be combined, and the exponents are simplified.

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 GCF
Not simplifying the exponents
Not combining like terms
Not checking the expression for any errors

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through practice problems, such as those found in a textbook or online resource.

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

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

Solving equations and inequalities
Graphing functions
Modeling real-world situations
Optimizing systems

Q: Can you provide some examples of simplifying algebraic expressions?

A: Here are some examples of simplifying algebraic expressions:

$\frac{2x^2y - 3}{6x^{-2}y^{-2}} = \frac{x^4y^3}{3}$
$\frac{4a^3b^2 - 2}{8a^{-1}b^{-1}} = \frac{a^4b^3}{2}$
$\frac{3c^2d - 2}{9c^{-2}d^{-2}} = \frac{c^4d^3}{3}$

Q: How can I use technology to simplify algebraic expressions?

A: You can use technology, such as a graphing calculator or computer algebra system, to simplify algebraic expressions.

Q: What are some tips for simplifying algebraic expressions?

A: Some tips for simplifying algebraic expressions include:

Start by factoring out the GCF
Simplify the exponents
Combine like terms
Check the expression for any errors
Use technology to simplify the expression if necessary

Conclusion

Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article, you can simplify even the most complex expressions. Remember to factor out the GCF, simplify the exponents, and simplify the numerator and denominator. With practice and patience, you will become proficient in simplifying algebraic expressions.