Find The Value Of The Below Algebraic Expression If A = 5 A = 5 A = 5 , B = 2 B = 2 B = 2 , And C = 4 C = 4 C = 4 : \frac{5}{12}\left(\frac{a^2-c^2}{a^2+b^2}\right ]

Mar 9, 2025 by ADMIN 166 views

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

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them requires a deep understanding of mathematical operations and formulas. In this article, we will focus on solving a specific algebraic expression involving variables $a$ , $b$ , and $c$ . We will substitute the given values of $a$ , $b$ , and $c$ into the expression and simplify it to find the final value.

The Algebraic Expression

The given algebraic expression is:

$\frac{5}{12}\left(\frac{a^2-c^2}{a^2+b^2}\right)$

We are given the values of $a$ , $b$ , and $c$ as $a = 5$ , $b = 2$ , and $c = 4$ . Our goal is to substitute these values into the expression and simplify it to find the final value.

Substituting Values into the Expression

To substitute the values of $a$ , $b$ , and $c$ into the expression, we will replace each variable with its corresponding value.

$\frac{5}{12}\left(\frac{a^2-c^2}{a^2+b^2}\right)$

becomes

$\frac{5}{12}\left(\frac{5^2-4^2}{5^2+2^2}\right)$

Simplifying the Expression

Now that we have substituted the values of $a$ , $b$ , and $c$ into the expression, we can simplify it by performing the necessary mathematical operations.

First, we will calculate the values of $5^2$ and $4^2$ .

$5^2 = 25$

$4^2 = 16$

Next, we will calculate the value of $5^2-4^2$ .

$5^2-4^2 = 25-16 = 9$

Then, we will calculate the value of $5^2+2^2$ .

$5^2+2^2 = 25+4 = 29$

Now, we can substitute these values back into the expression.

$\frac{5}{12}\left(\frac{5^2-4^2}{5^2+2^2}\right)$

becomes

$\frac{5}{12}\left(\frac{9}{29}\right)$

Final Simplification

To simplify the expression further, we can multiply the numerator and denominator by the same value to eliminate the fraction in the numerator.

$\frac{5}{12}\left(\frac{9}{29}\right)$

becomes

$\frac{5 \times 9}{12 \times 29}$

Calculating the Final Value

Now that we have simplified the expression, we can calculate the final value by performing the necessary mathematical operations.

$\frac{5 \times 9}{12 \times 29}$

becomes

$\frac{45}{348}$

To simplify this fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 3.

$\frac{45}{348}$

becomes

$\frac{15}{116}$

Conclusion

In this article, we have solved a specific algebraic expression involving variables $a$ , $b$ , and $c$ . We substituted the given values of $a$ , $b$ , and $c$ into the expression and simplified it to find the final value. The final value of the expression is $\frac{15}{116}$ . This example demonstrates the importance of understanding mathematical operations and formulas in solving algebraic expressions.

Tips and Tricks

When solving algebraic expressions, it is essential to follow the order of operations (PEMDAS):

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.

By following this order of operations, you can ensure that you are solving algebraic expressions correctly and efficiently.

Common Mistakes to Avoid

When solving algebraic expressions, there are several common mistakes to avoid:

Incorrect substitution: Make sure to substitute the correct values of variables into the expression.
Incorrect simplification: Make sure to simplify the expression correctly by following the order of operations.
Incorrect calculation: Make sure to perform the necessary mathematical operations correctly.

By avoiding these common mistakes, you can ensure that you are solving algebraic expressions accurately and efficiently.

Real-World Applications

Algebraic expressions have numerous real-world applications in various fields, including:

Science: Algebraic expressions are used to model and analyze scientific data.
Engineering: Algebraic expressions are used to design and optimize engineering systems.
Finance: Algebraic expressions are used to model and analyze financial data.

By understanding algebraic expressions, you can apply mathematical concepts to real-world problems and make informed decisions.

Conclusion

In conclusion, solving algebraic expressions requires a deep understanding of mathematical operations and formulas. By following the order of operations and avoiding common mistakes, you can ensure that you are solving algebraic expressions accurately and efficiently. The final value of the expression is $\frac{15}{116}$ . This example demonstrates the importance of understanding mathematical operations and formulas in solving algebraic expressions.

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them requires a deep understanding of mathematical operations and formulas. In this article, we will answer some frequently asked questions about algebraic expressions, covering topics such as substitution, simplification, and real-world applications.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations. Algebraic expressions are used to model and analyze real-world problems, and they are a fundamental concept in mathematics.

Q: How do I substitute values into an algebraic expression?

A: To substitute values into an algebraic expression, you need to replace each variable with its corresponding value. For example, if you have the expression $a^2 + b^2$ and you want to substitute $a = 5$ and $b = 2$ , you would replace $a$ and $b$ with their corresponding values to get $5^2 + 2^2$ .

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to follow the order of operations (PEMDAS):

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.

By following this order of operations, you can simplify algebraic expressions correctly and efficiently.

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

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

Incorrect substitution: Make sure to substitute the correct values of variables into the expression.
Incorrect simplification: Make sure to simplify the expression correctly by following the order of operations.
Incorrect calculation: Make sure to perform the necessary mathematical operations correctly.

By avoiding these common mistakes, you can ensure that you are solving algebraic expressions accurately and efficiently.

Q: How do algebraic expressions have real-world applications?

A: Algebraic expressions have numerous real-world applications in various fields, including:

Science: Algebraic expressions are used to model and analyze scientific data.
Engineering: Algebraic expressions are used to design and optimize engineering systems.
Finance: Algebraic expressions are used to model and analyze financial data.

By understanding algebraic expressions, you can apply mathematical concepts to real-world problems and make informed decisions.

Q: What are some tips for solving algebraic expressions?

A: Some tips for solving algebraic expressions include:

Follow the order of operations: Make sure to follow the order of operations (PEMDAS) to simplify algebraic expressions correctly.
Use substitution: Use substitution to replace variables with their corresponding values.
Simplify carefully: Simplify algebraic expressions carefully to avoid errors.

By following these tips, you can solve algebraic expressions accurately and efficiently.

Q: How can I practice solving algebraic expressions?

A: You can practice solving algebraic expressions by:

Solving problems: Solve algebraic expression problems to practice your skills.
Using online resources: Use online resources, such as algebraic expression calculators and worksheets, to practice solving algebraic expressions.
Working with a tutor: Work with a tutor or teacher to practice solving algebraic expressions.

By practicing solving algebraic expressions, you can improve your skills and become more confident in your ability to solve them.

Conclusion

In conclusion, algebraic expressions are a fundamental concept in mathematics, and solving them requires a deep understanding of mathematical operations and formulas. By following the order of operations, avoiding common mistakes, and practicing solving algebraic expressions, you can become more confident in your ability to solve them. Remember to use substitution, simplify carefully, and follow the order of operations to solve algebraic expressions accurately and efficiently.