Simplify The Expression:$\[ \frac{15 A^2 B^2}{21 A C} \cdot \frac{14 A^4 C^2}{6 A B^3} \\]

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Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying a given expression involving variables and constants. We will break down the problem into manageable steps, and by the end of this guide, you will be able to simplify the expression with ease.

The Given Expression

The given expression is:

15a2b221acâ‹…14a4c26ab3\frac{15 a^2 b^2}{21 a c} \cdot \frac{14 a^4 c^2}{6 a b^3}

Step 1: Factor Out Common Terms

To simplify the expression, we need to factor out common terms from both the numerator and the denominator. Let's start by factoring out the common terms from the first fraction:

15a2b221ac=5ab27c\frac{15 a^2 b^2}{21 a c} = \frac{5 a b^2}{7 c}

Similarly, let's factor out the common terms from the second fraction:

14a4c26ab3=7a3c23b3\frac{14 a^4 c^2}{6 a b^3} = \frac{7 a^3 c^2}{3 b^3}

Step 2: Multiply the Fractions

Now that we have factored out the common terms, we can multiply the fractions together:

5ab27câ‹…7a3c23b3\frac{5 a b^2}{7 c} \cdot \frac{7 a^3 c^2}{3 b^3}

Step 3: Simplify the Expression

When multiplying fractions, we can simply multiply the numerators together and the denominators together:

5ab2â‹…7a3c27câ‹…3b3\frac{5 a b^2 \cdot 7 a^3 c^2}{7 c \cdot 3 b^3}

Step 4: Cancel Out Common Terms

Now that we have multiplied the fractions, we can cancel out common terms between the numerator and the denominator:

5ab2â‹…7a3c27câ‹…3b3=5a4b2c23b3c\frac{5 a b^2 \cdot 7 a^3 c^2}{7 c \cdot 3 b^3} = \frac{5 a^4 b^2 c^2}{3 b^3 c}

Step 5: Simplify Further

We can simplify the expression further by canceling out common terms between the numerator and the denominator:

5a4b2c23b3c=5a4c3bc\frac{5 a^4 b^2 c^2}{3 b^3 c} = \frac{5 a^4 c}{3 b c}

Step 6: Final Simplification

The final step is to simplify the expression by canceling out any remaining common terms:

5a4c3bc=5a33b\frac{5 a^4 c}{3 b c} = \frac{5 a^3}{3 b}

Conclusion

In this article, we have simplified the given expression step by step. We have factored out common terms, multiplied the fractions, canceled out common terms, and finally simplified the expression to its simplest form. By following these steps, you can simplify any algebraic expression with ease.

Frequently Asked Questions

Q: What is the final simplified expression?

A: The final simplified expression is 5a33b\frac{5 a^3}{3 b}.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to factor out common terms, multiply the fractions, cancel out common terms, and finally simplify the expression to its simplest form.

Q: What are the common terms that can be canceled out?

A: Common terms that can be canceled out include variables and constants that appear in both the numerator and the denominator.

Final Answer

The final answer is 5a33b\boxed{\frac{5 a^3}{3 b}}.

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Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will provide a comprehensive Q&A guide to algebraic manipulation, covering various topics and concepts.

Q: What is Algebraic Manipulation?

A: Algebraic manipulation is the process of simplifying and rearranging algebraic expressions to make them easier to work with. It involves using various techniques such as factoring, multiplying, and canceling out common terms to simplify expressions.

Q: What are the Basic Steps of Algebraic Manipulation?

A: The basic steps of algebraic manipulation include:

  1. Factoring out common terms
  2. Multiplying fractions
  3. Canceling out common terms
  4. Simplifying the expression to its simplest form

Q: How Do I Factor Out Common Terms?

A: To factor out common terms, you need to identify the common factors in the numerator and the denominator. You can then group the common factors together and write them as a single term.

Q: What is the Difference Between Factoring and Canceling Out Common Terms?

A: Factoring involves grouping common factors together and writing them as a single term, while canceling out common terms involves eliminating common factors between the numerator and the denominator.

Q: How Do I Multiply Fractions?

A: To multiply fractions, you need to multiply the numerators together and the denominators together. You can then simplify the resulting fraction by canceling out common terms.

Q: What is the Order of Operations in Algebraic Manipulation?

A: The order of operations in algebraic manipulation is:

  1. Parentheses: Evaluate expressions inside parentheses first
  2. Exponents: Evaluate any exponential expressions next
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right

Q: How Do I Simplify an Algebraic Expression?

A: To simplify an algebraic expression, you need to follow the basic steps of algebraic manipulation:

  1. Factor out common terms
  2. Multiply fractions
  3. Cancel out common terms
  4. Simplify the expression to its simplest form

Q: What are Some Common Algebraic Manipulation Techniques?

A: Some common algebraic manipulation techniques include:

  1. Factoring: Factoring out common terms
  2. Canceling Out Common Terms: Eliminating common factors between the numerator and the denominator
  3. Multiplying Fractions: Multiplying the numerators together and the denominators together
  4. Simplifying Expressions: Simplifying expressions to their simplest form

Q: How Do I Use Algebraic Manipulation in Real-World Applications?

A: Algebraic manipulation is used in various real-world applications, including:

  1. Physics: Algebraic manipulation is used to solve problems involving motion, energy, and forces
  2. Engineering: Algebraic manipulation is used to design and optimize systems, such as electrical circuits and mechanical systems
  3. Economics: Algebraic manipulation is used to model and analyze economic systems, including supply and demand curves
  4. Computer Science: Algebraic manipulation is used in computer science to solve problems involving algorithms and data structures

Conclusion

In this article, we have provided a comprehensive Q&A guide to algebraic manipulation, covering various topics and concepts. We hope that this guide has been helpful in understanding the basics of algebraic manipulation and how to apply it in real-world applications.

Frequently Asked Questions

Q: What is the final simplified expression?

A: The final simplified expression depends on the specific problem being solved.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to follow the basic steps of algebraic manipulation: factoring out common terms, multiplying fractions, canceling out common terms, and simplifying the expression to its simplest form.

Q: What are some common algebraic manipulation techniques?

A: Some common algebraic manipulation techniques include factoring, canceling out common terms, multiplying fractions, and simplifying expressions.

Final Answer

The final answer is 5a33b\boxed{\frac{5 a^3}{3 b}}.