Simplify The Expression: $\frac{1}{2} A^{x-3} B^{2n} \div -\frac{3}{4} A^{-2} B^{3n-1}$

Feb 28, 2025 by ADMIN 88 views

Simplify the Expression: A Comprehensive Guide to Algebraic Manipulation

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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 delve into the world of algebraic manipulation and explore the step-by-step process of simplifying a complex expression. Our focus will be on the expression $\frac{1}{2} a^{x-3} b^{2n} \div -\frac{3}{4} a^{-2} b^{3n-1}$ , and we will break down the simplification process into manageable chunks.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its components. The expression consists of two fractions, $\frac{1}{2}$ and $-\frac{3}{4}$ , multiplied by variables $a$ and $b$ raised to certain powers. The expression is divided by the second fraction, which means we will be performing a division operation.

Variables and Exponents

In algebra, variables are represented by letters, and exponents are used to indicate the power to which a variable is raised. In this expression, we have two variables, $a$ and $b$ , and their respective exponents are $x-3$ and $2n$ for $a$ , and $3n-1$ for $b$ . The negative exponent $-2$ indicates that the variable $a$ is raised to the power of $-2$ .

Division of Fractions

When dividing fractions, we need to invert the second fraction and multiply. This means that we will multiply the first fraction by the reciprocal of the second fraction.

Step 1: Invert the Second Fraction

To simplify the expression, we need to invert the second fraction, which means changing the sign of the numerator and denominator. The inverted fraction becomes $\frac{4}{3} a^{2} b^{1-3n}$ .

Step 2: Multiply the Fractions

Now that we have inverted the second fraction, we can multiply the two fractions together. This means multiplying the numerators and denominators separately.

Multiplying Numerators

The numerators are $\frac{1}{2}$ and $\frac{4}{3}$ . Multiplying these two fractions gives us $\frac{1 \times 4}{2 \times 3} = \frac{4}{6}$ .

Multiplying Denominators

The denominators are $a^{x-3} b^{2n}$ and $a^{2} b^{1-3n}$ . Multiplying these two expressions gives us $a^{x-3+2} b^{2n+1-3n}$ .

Step 3: Simplify the Expression

Now that we have multiplied the fractions, we can simplify the expression by combining like terms.

Combining Like Terms

The expression $a^{x-3+2} b^{2n+1-3n}$ can be simplified by combining the exponents. This gives us $a^{x-1} b^{2n-2n+1}$ , which simplifies to $a^{x-1} b^{1}$ .

Step 4: Simplify the Numerator

The numerator of the expression is $\frac{4}{6}$ . This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2.

Simplifying the Numerator

Dividing both the numerator and denominator by 2 gives us $\frac{2}{3}$ .

Step 5: Write the Final Expression

Now that we have simplified the expression, we can write the final result.

Final Expression

The final expression is $\frac{2}{3} a^{x-1} b^{1}$ .

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the step-by-step process outlined in this article, we have successfully simplified the expression $\frac{1}{2} a^{x-3} b^{2n} \div -\frac{3}{4} a^{-2} b^{3n-1}$ . We have broken down the simplification process into manageable chunks, and we have used algebraic manipulation to simplify the expression.

Key Takeaways

Algebraic expressions can be simplified by following a step-by-step process.
Division of fractions involves inverting the second fraction and multiplying.
Exponents can be combined by adding or subtracting the exponents.
Like terms can be combined by adding or subtracting the coefficients.

Final Thoughts

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the step-by-step process outlined in this article, we have successfully simplified the expression $\frac{1}{2} a^{x-3} b^{2n} \div -\frac{3}{4} a^{-2} b^{3n-1}$ . We hope that this article has provided valuable insights into the world of algebraic manipulation and has inspired readers to explore the world of mathematics.

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Introduction

In our previous article, we explored the step-by-step process of simplifying a complex algebraic expression. We broke down the expression $\frac{1}{2} a^{x-3} b^{2n} \div -\frac{3}{4} a^{-2} b^{3n-1}$ into manageable chunks and used algebraic manipulation to simplify it. In this article, we will answer some of the most frequently asked questions related to 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 identify the components of the expression, including the variables, exponents, and fractions.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.

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.

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

A: To simplify an expression with negative exponents, you need to rewrite the expression with positive exponents by moving the variable to the other side of the fraction.

Q: What is the rule for multiplying exponents?

A: When multiplying exponents, you add the exponents together.

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

A: To simplify an expression with multiple fractions, you need to multiply the numerators and denominators separately and then simplify the resulting fraction.

Q: What is the rule for dividing fractions?

A: When dividing fractions, you need to invert the second fraction and multiply.

Q: How do I simplify an expression with variables in the denominator?

A: To simplify an expression with variables in the denominator, you need to multiply the numerator and denominator by the reciprocal of the variable.

Q: What is the final expression after simplifying the given expression?

A: The final expression after simplifying the given expression is $\frac{2}{3} a^{x-1} b^{1}$ .

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the step-by-step process outlined in this article, we have successfully simplified the expression $\frac{1}{2} a^{x-3} b^{2n} \div -\frac{3}{4} a^{-2} b^{3n-1}$ . We hope that this article has provided valuable insights into the world of algebraic manipulation and has inspired readers to explore the world of mathematics.

Key Takeaways

Algebraic expressions can be simplified by following a step-by-step process.
Division of fractions involves inverting the second fraction and multiplying.
Exponents can be combined by adding or subtracting the exponents.
Like terms can be combined by adding or subtracting the coefficients.

Final Thoughts

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the step-by-step process outlined in this article, we have successfully simplified the expression $\frac{1}{2} a^{x-3} b^{2n} \div -\frac{3}{4} a^{-2} b^{3n-1}$ . We hope that this article has provided valuable insights into the world of algebraic manipulation and has inspired readers to explore the world of mathematics.

Additional Resources

For more information on simplifying algebraic expressions, we recommend the following resources:

Khan Academy: Algebraic Manipulation
Mathway: Algebraic Manipulation
Wolfram Alpha: Algebraic Manipulation

We hope that this article has provided valuable insights into the world of algebraic manipulation and has inspired readers to explore the world of mathematics.