9. Simplify The Expression: $\frac{a+b}{c}$

Mar 1, 2025 by ADMIN 44 views

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

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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 focus on simplifying the expression $\frac{a+b}{c}$ , which is a common algebraic expression that can be simplified using various techniques. We will break down the process into step-by-step instructions, making it easy to understand and apply.

Understanding the Expression

The given expression is $\frac{a+b}{c}$ . This expression consists of two terms in the numerator, $a$ and $b$ , which are added together, and a single term in the denominator, $c$ . The goal is to simplify this expression by combining like terms and performing any necessary operations.

Step 1: Identify Like Terms

Like terms are terms that have the same variable raised to the same power. In this expression, the only like terms are $a$ and $b$ , which are both variables raised to the power of 1. To simplify the expression, we need to combine these like terms.

Step 2: Combine Like Terms

To combine like terms, we add their coefficients. In this case, the coefficients are 1 for both $a$ and $b$ . Therefore, we can combine them as follows:

\frac{a+b}{c} = \frac{1a + 1b}{c} = \frac{a+b}{c}

Step 3: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by canceling out any common factors. In this case, there are no common factors between the numerator and the denominator, so the expression remains the same.

Step 4: Final Answer

The final answer is $\frac{a+b}{c}$ .

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the step-by-step instructions outlined in this article, you can simplify the expression $\frac{a+b}{c}$ and arrive at the final answer. Remember to identify like terms, combine them, and simplify the expression by canceling out any common factors.

Tips and Variations

To simplify more complex expressions, you can use the distributive property to expand the numerator and then combine like terms.
You can also use the commutative property to rearrange the terms in the numerator and denominator.
If the expression has a common factor between the numerator and denominator, you can cancel it out to simplify the expression.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications in various fields, including:

Science: Algebraic expressions are used to model real-world phenomena, such as population growth and chemical reactions.
Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
Finance: Algebraic expressions are used to calculate interest rates and investment returns.

Common Mistakes

When simplifying algebraic expressions, it's essential to avoid common mistakes, such as:

Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.
Not canceling out common factors: Failing to cancel out common factors can lead to incorrect simplifications.
Not following the order of operations: Failing to follow the order of operations can lead to incorrect simplifications.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

Simplify the expression $\frac{2a+3b}{4c}$ .
Simplify the expression $\frac{a-b}{c+d}$ .
Simplify the expression $\frac{2a+3b-4c}{5d}$ .

Conclusion

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Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify like terms. Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you add their coefficients. For example, if you have the expression $2a + 3a$ , you can combine the like terms by adding their coefficients: $2a + 3a = 5a$ .

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

A: A coefficient is a number that is multiplied by a variable. For example, in the expression $2a$ , the coefficient is 2 and the variable is $a$ . A variable is a letter or symbol that represents a value that can change.

Q: Can I simplify an expression by canceling out common factors?

A: Yes, you can simplify an expression by canceling out common factors. For example, if you have the expression $\frac{2a}{4a}$ , you can cancel out the common factor of $2a$ by dividing both the numerator and the denominator by $2a$ : $\frac{2a}{4a} = \frac{1}{2}$ .

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an expression. The order of operations is:

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 simplify an expression with fractions?

A: To simplify an expression with fractions, you can follow these steps:

Simplify the numerator and denominator separately.
Look for any common factors between the numerator and denominator.
Cancel out any common factors.
Simplify the resulting expression.

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

A: Yes, you can simplify an expression with variables in the denominator. However, you must be careful not to divide by zero. If the denominator is a variable, you can simplify the expression by canceling out any common factors between the numerator and denominator.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can follow these steps:

Simplify the expression inside the exponent.
Evaluate any exponential expressions.
Simplify the resulting expression.

Q: Can I simplify an expression with absolute values?

A: Yes, you can simplify an expression with absolute values. However, you must be careful when simplifying expressions with absolute values, as the absolute value of a negative number is always positive.

Q: How do I simplify an expression with radicals?

A: To simplify an expression with radicals, you can follow these steps:

Simplify the expression inside the radical.
Evaluate any radical expressions.
Simplify the resulting expression.

Q: Can I simplify an expression with trigonometric functions?

A: Yes, you can simplify an expression with trigonometric functions. However, you must be careful when simplifying expressions with trigonometric functions, as the trigonometric functions can be complex and require careful evaluation.

Q: How do I simplify an expression with logarithmic functions?

A: To simplify an expression with logarithmic functions, you can follow these steps:

Simplify the expression inside the logarithm.
Evaluate any logarithmic expressions.
Simplify the resulting expression.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the step-by-step instructions outlined in this article, you can simplify a wide range of expressions, from simple fractions to complex trigonometric functions. Remember to identify like terms, combine them, and simplify the expression by canceling out any common factors. With practice and patience, you can become proficient in simplifying algebraic expressions and apply them to real-world problems.