Simplify The Expression:${ \frac{\left(2 A^5 B^3\right)\left(2 A^2 B^4\right)}{32 A^{19} B^{17}} }$

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Introduction

Algebraic expressions can be complex and daunting, especially when dealing with exponents and variables. However, with a clear understanding of the rules and techniques involved, simplifying these expressions becomes a manageable task. In this article, we will focus on simplifying a specific expression involving exponents and variables, and provide a step-by-step guide on how to approach it.

The Expression to Simplify

The given expression is:

(2a5b3)(2a2b4)32a19b17\frac{\left(2 a^5 b^3\right)\left(2 a^2 b^4\right)}{32 a^{19} b^{17}}

This expression involves multiplication and division of terms with exponents, making it a good candidate for simplification.

Step 1: Multiply the Numerators

To simplify the expression, we start by multiplying the numerators:

(2a5b3)(2a2b4)\left(2 a^5 b^3\right)\left(2 a^2 b^4\right)

Using the rule for multiplying exponents with the same base, we add the exponents:

2⋅2⋅a5+2⋅b3+42 \cdot 2 \cdot a^{5+2} \cdot b^{3+4}

=4a7b7= 4 a^7 b^7

So, the numerator simplifies to 4a7b74 a^7 b^7.

Step 2: Simplify the Denominator

Next, we simplify the denominator:

32a19b1732 a^{19} b^{17}

We can rewrite 3232 as 252^5, and then use the rule for dividing exponents with the same base:

25a19b171\frac{2^5 a^{19} b^{17}}{1}

=25a19b17= 2^5 a^{19} b^{17}

However, we can simplify this further by expressing 252^5 as 3232:

=32a19b17= 32 a^{19} b^{17}

Step 3: Divide the Numerator by the Denominator

Now that we have simplified the numerator and denominator, we can divide the numerator by the denominator:

4a7b732a19b17\frac{4 a^7 b^7}{32 a^{19} b^{17}}

Using the rule for dividing exponents with the same base, we subtract the exponents:

=432⋅a7−19⋅b7−17= \frac{4}{32} \cdot a^{7-19} \cdot b^{7-17}

=18⋅a−12⋅b−10= \frac{1}{8} \cdot a^{-12} \cdot b^{-10}

Step 4: Simplify the Result

Finally, we can simplify the result by expressing the fraction as a product of powers of the variables:

=18⋅a−12⋅b−10= \frac{1}{8} \cdot a^{-12} \cdot b^{-10}

=a−12b−108= \frac{a^{-12} b^{-10}}{8}

=18⋅1a12b10= \frac{1}{8} \cdot \frac{1}{a^{12} b^{10}}

=18a12b10= \frac{1}{8 a^{12} b^{10}}

Conclusion

Simplifying complex algebraic expressions requires a clear understanding of the rules and techniques involved. By following the steps outlined in this article, we can simplify expressions involving exponents and variables. Remember to multiply the numerators, simplify the denominator, divide the numerator by the denominator, and finally simplify the result.

Common Mistakes to Avoid

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

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
  • Not using the correct rules for exponents: Remember to use the correct rules for multiplying and dividing exponents with the same base.
  • Not simplifying the denominator: Don't forget to simplify the denominator before dividing the numerator by it.
  • Not checking the final result: Always check the final result to make sure it's simplified and correct.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. For example:

  • Science and Engineering: Simplifying expressions is crucial in science and engineering, where complex equations are used to model real-world phenomena.
  • Finance: Simplifying expressions is also important in finance, where complex financial models are used to predict stock prices and investment returns.
  • Computer Science: Simplifying expressions is a fundamental concept in computer science, where algorithms and data structures are used to solve complex problems.

Final Thoughts

Introduction

Simplifying algebraic expressions can be a challenging task, especially for those who are new to algebra. However, with practice and patience, it becomes a manageable task. In this article, we will provide a Q&A guide to help you simplify algebraic expressions and develop a deeper understanding of algebraic concepts.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: What are the rules for simplifying algebraic expressions?

A: The rules for simplifying algebraic expressions are:

  • Combine like terms: Combine terms that have the same variable and exponent.
  • Simplify fractions: Simplify fractions by dividing the numerator and denominator by their greatest common divisor.
  • Use the order of operations: Follow the order of operations (PEMDAS) when simplifying expressions.
  • Use the rules for exponents: Use the rules for multiplying and dividing exponents with the same base.

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

A: To simplify an expression with multiple variables, follow these steps:

  • Identify the variables: Identify the variables in the expression and their corresponding exponents.
  • Combine like terms: Combine terms that have the same variable and exponent.
  • Simplify fractions: Simplify fractions by dividing the numerator and denominator by their greatest common divisor.
  • Use the order of operations: Follow the order of operations (PEMDAS) when simplifying expressions.

Q: What is the difference between a variable and a constant?

A: A variable is a 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 a negative exponent?

A: To simplify an expression with a negative exponent, follow these steps:

  • Identify the negative exponent: Identify the negative exponent in the expression.
  • Rewrite the expression: Rewrite the expression with a positive exponent by moving the variable to the other side of the fraction.
  • Simplify the expression: Simplify the expression by combining like terms and simplifying fractions.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be simplified to a fraction, while an irrational expression is an expression that cannot be simplified to a fraction.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, follow these steps:

  • Factor the numerator and denominator: Factor the numerator and denominator to identify any common factors.
  • Cancel out common factors: Cancel out any common factors between the numerator and denominator.
  • Simplify the expression: Simplify the expression by combining like terms and simplifying fractions.

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

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

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
  • Not using the correct rules for exponents: Remember to use the correct rules for multiplying and dividing exponents with the same base.
  • Not simplifying the denominator: Don't forget to simplify the denominator before dividing the numerator by it.
  • Not checking the final result: Always check the final result to make sure it's simplified and correct.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and with practice and patience, it becomes a manageable task. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex expressions and develop a deeper understanding of algebraic concepts. Remember to always check your work and seek help when needed. With time and practice, you'll become proficient in simplifying algebraic expressions and be able to tackle even the most complex problems.