Perform The Indicated Operation. 14 A 2 10 B 2 ÷ 21 A 2 15 B 2 = \frac{14 A^2}{10 B^2} \div \frac{21 A^2}{15 B^2} = 10 B 2 14 A 2 ÷ 15 B 2 21 A 2 =

Mar 9, 2025 by ADMIN 152 views

**Performing Indicated Operations in Algebra: A Step-by-Step Guide**

Introduction

In algebra, performing indicated operations is a crucial skill that helps us simplify complex expressions and solve equations. In this article, we will focus on performing division operations involving fractions with variables. We will use the given problem $\frac{14 a^2}{10 b^2} \div \frac{21 a^2}{15 b^2}$ as a case study to demonstrate the step-by-step process.

Understanding the Problem

The given problem involves dividing two fractions with variables. To solve this problem, we need to follow the order of operations (PEMDAS) and apply the rules of fraction division.

Step 1: Invert the Divisor

When dividing fractions, we need to invert the divisor, which means flipping the numerator and denominator of the second fraction. In this case, we will invert $\frac{21 a^2}{15 b^2}$ to get $\frac{15 b^2}{21 a^2}$ .

$\frac{14 a^2}{10 b^2} \div \frac{21 a^2}{15 b^2} = \frac{14 a^2}{10 b^2} \times \frac{15 b^2}{21 a^2}$

Step 2: Multiply the Numerators and Denominators

Now that we have inverted the divisor, we can multiply the numerators and denominators of the two fractions. This will give us a new fraction with a numerator and denominator that are the product of the original numerators and denominators.

$\frac{14 a^2}{10 b^2} \times \frac{15 b^2}{21 a^2} = \frac{(14 a^2) \times (15 b^2)}{(10 b^2) \times (21 a^2)}$

Step 3: Simplify the Fraction

Now that we have multiplied the numerators and denominators, we can simplify the fraction by canceling out any common factors. In this case, we can cancel out the $a^2$ terms in the numerator and denominator, as well as the $b^2$ terms.

$\frac{(14 a^2) \times (15 b^2)}{(10 b^2) \times (21 a^2)} = \frac{14 \times 15}{10 \times 21}$

Step 4: Evaluate the Expression

Now that we have simplified the fraction, we can evaluate the expression by performing the multiplication and division operations.

$\frac{14 \times 15}{10 \times 21} = \frac{210}{210}$

Conclusion

In this article, we have demonstrated how to perform indicated operations in algebra, specifically division operations involving fractions with variables. We have used the given problem $\frac{14 a^2}{10 b^2} \div \frac{21 a^2}{15 b^2}$ as a case study to illustrate the step-by-step process. By following the order of operations and applying the rules of fraction division, we have simplified the expression and evaluated the result.

Final Answer

The final answer to the given problem is $\boxed{\frac{210}{210}}$ , which simplifies to $\boxed{1}$ .

Common Mistakes to Avoid

When performing indicated operations in algebra, it's essential to avoid common mistakes such as:

Forgetting to invert the divisor when dividing fractions
Failing to multiply the numerators and denominators correctly
Not simplifying the fraction by canceling out common factors
Not evaluating the expression correctly

By following the step-by-step process outlined in this article and avoiding common mistakes, you can become proficient in performing indicated operations in algebra and solve complex problems with confidence.

Real-World Applications

Performing indicated operations in algebra has numerous real-world applications, including:

Science and Engineering: Algebraic expressions are used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits.
Computer Programming: Algebraic expressions are used to write algorithms and solve problems in computer programming.
Economics: Algebraic expressions are used to model economic systems, such as supply and demand curves, and to solve problems in finance and accounting.

By mastering the skills outlined in this article, you can apply algebraic techniques to solve real-world problems and make informed decisions in various fields.

Conclusion

Introduction

In our previous article, we demonstrated how to perform indicated operations in algebra, specifically division operations involving fractions with variables. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques outlined in the previous article.

Q: What is the order of operations in algebra?

A: The order of operations in algebra is PEMDAS, which stands for:

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

Q: How do I invert the divisor when dividing fractions?

A: To invert the divisor when dividing fractions, you need to flip the numerator and denominator of the second fraction. For example, if you are dividing $\frac{a}{b}$ by $\frac{c}{d}$ , you would invert the second fraction to get $\frac{d}{c}$ .

Q: What is the rule for multiplying fractions?

A: The rule for multiplying fractions is to multiply the numerators and denominators separately. For example, if you are multiplying $\frac{a}{b}$ by $\frac{c}{d}$ , you would multiply the numerators to get $ac$ and the denominators to get $bd$ .

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to cancel out any common factors between the numerator and denominator. For example, if you have the fraction $\frac{6}{8}$ , you can simplify it by canceling out the common factor of 2 to get $\frac{3}{4}$ .

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. For example, in the expression $x + 5$ , $x$ is a variable and $5$ is a constant.

Q: How do I evaluate an expression with variables?

A: To evaluate an expression with variables, you need to substitute the value of the variable into the expression. For example, if you have the expression $x + 5$ and $x = 3$ , you would substitute $3$ into the expression to get $3 + 5 = 8$ .

Q: What are some common mistakes to avoid when performing indicated operations in algebra?

A: Some common mistakes to avoid when performing indicated operations in algebra include:

Forgetting to invert the divisor when dividing fractions
Failing to multiply the numerators and denominators correctly
Not simplifying the fraction by canceling out common factors
Not evaluating the expression correctly

Q: How do I apply algebraic techniques to real-world problems?

A: Algebraic techniques can be applied to real-world problems in various fields, including science, engineering, computer programming, and economics. For example, algebraic expressions can be used to model population growth, chemical reactions, and electrical circuits.

Conclusion

In conclusion, performing indicated operations in algebra is a crucial skill that helps us simplify complex expressions and solve equations. By following the step-by-step process outlined in this article and avoiding common mistakes, you can become proficient in performing indicated operations in algebra and solve complex problems with confidence.

Final Tips

Practice, practice, practice: The more you practice performing indicated operations in algebra, the more comfortable you will become with the concepts and techniques.
Use online resources: There are many online resources available that can help you learn and practice algebra, including video tutorials, practice problems, and interactive simulations.
Seek help when needed: Don't be afraid to ask for help if you are struggling with a concept or technique. Your teacher, tutor, or classmate may be able to provide additional support and guidance.

By following these tips and practicing regularly, you can become proficient in performing indicated operations in algebra and solve complex problems with confidence.