Exercise 91. Simplify The Following: A) $\frac{2x}{3} + \frac{1}{6}$ B) $\frac{3}{x} + \frac{2}{x^2}$ C) $\frac{3}{2x^2y} - \frac{5}{4xy} + \frac{7}{6x} - 4$2. Simplify The Following: A) $\frac{x-3}{3} -

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 explore the process of simplifying three different types of algebraic expressions, focusing on fractions and mixed numbers. We will break down each expression into manageable steps, making it easier to understand and apply the concepts.

Simplifying Fractions: A Case Study

a) $\frac{2x}{3} + \frac{1}{6}$

To simplify this expression, we need to find a common denominator. The least common multiple (LCM) of 3 and 6 is 6. We can rewrite the first fraction with a denominator of 6:

$$\frac{2x}{3} = \frac{2x \cdot 2}{3 \cdot 2} = \frac{4x}{6}$$

Now we can add the two fractions:

$$\frac{4x}{6} + \frac{1}{6} = \frac{4x + 1}{6}$$

This is the simplified form of the expression.

b) $\frac{3}{x} + \frac{2}{x^2}$

To simplify this expression, we need to find a common denominator. The least common multiple (LCM) of x and x^2 is x^2. We can rewrite the first fraction with a denominator of x^2:

$$\frac{3}{x} = \frac{3 \cdot x}{x \cdot x} = \frac{3x}{x^2}$$

Now we can add the two fractions:

$$\frac{3x}{x^2} + \frac{2}{x^2} = \frac{3x + 2}{x^2}$$

This is the simplified form of the expression.

c) $\frac{3}{2x^2y} - \frac{5}{4xy} + \frac{7}{6x} - 4$

To simplify this expression, we need to find a common denominator. The least common multiple (LCM) of 2x^2y, 4xy, 6x, and 1 is 12x^2y. We can rewrite each fraction with a denominator of 12x^2y:

$$\frac{3}{2x^2y} = \frac{3 \cdot 6}{2 \cdot 6 \cdot x^2y} = \frac{18}{12x^2y}$$

$$\frac{5}{4xy} = \frac{5 \cdot 3}{4 \cdot 3 \cdot xy} = \frac{15}{12xy}$$

$$\frac{7}{6x} = \frac{7 \cdot 2}{6 \cdot 2 \cdot x} = \frac{14}{12x}$$

Now we can rewrite the expression with a common denominator:

$$\frac{18}{12x^2y} - \frac{15}{12xy} + \frac{14}{12x} - 4$$

We can combine the fractions:

$$\frac{18 - 15 \cdot x + 14 \cdot x^2y - 4 \cdot 12x^2y}{12x^2y}$$

Simplifying the numerator:

$$\frac{14x^2y - 15x + 18}{12x^2y}$$

This is the simplified form of the expression.

Simplifying Mixed Numbers: A Case Study

a) $\frac{x-3}{3} - 2$

To simplify this expression, we need to rewrite the mixed number as an improper fraction:

$$\frac{x-3}{3} = \frac{x-3}{3} \cdot \frac{1}{1} = \frac{x-3}{3}$$

Now we can subtract 2:

$$\frac{x-3}{3} - 2 = \frac{x-3}{3} - \frac{2 \cdot 3}{3} = \frac{x-3 - 6}{3}$$

Simplifying the numerator:

$$\frac{x-9}{3}$$

This is the simplified form of the expression.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify fractions and mixed numbers with ease. Remember to find a common denominator, rewrite fractions with a common denominator, and combine fractions. With practice, you will become proficient in simplifying algebraic expressions and be able to tackle even the most complex problems.

Tips and Tricks

• Always find a common denominator when adding or subtracting fractions.
• Rewrite fractions with a common denominator to make it easier to combine them.
• Simplify the numerator after combining fractions.
• Practice, practice, practice! The more you practice, the more comfortable you will become with simplifying algebraic expressions.

Common Mistakes to Avoid

• Failing to find a common denominator when adding or subtracting fractions.
• Not rewriting fractions with a common denominator.
• Not simplifying the numerator after combining fractions.
• Not practicing regularly to develop your skills.

Real-World Applications

Simplifying algebraic expressions has many real-world applications, including:

• Calculating interest rates and investments
• Determining the area and perimeter of shapes
• Solving problems in physics and engineering
• Creating models for population growth and other real-world phenomena

Introduction

Simplifying algebraic expressions is an essential skill for any math enthusiast. In our previous article, we explored the process of simplifying fractions and mixed numbers. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q: What is the difference between simplifying and evaluating an algebraic expression?

A: Simplifying an algebraic expression involves rewriting it in a more compact form, often by combining like terms or canceling out common factors. Evaluating an algebraic expression, on the other hand, involves substituting specific values for the variables and calculating the resulting value.

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression whenever possible, as it can make the expression easier to work with and understand. Simplifying an expression can also help you to identify patterns and relationships between the variables.

Q: What is the least common multiple (LCM) and how do I find it?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that appears in both lists.

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

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

1. Find a common denominator for all the fractions.
2. Rewrite each fraction with the common denominator.
3. Combine the fractions by adding or subtracting their numerators.
4. Simplify the resulting fraction, if possible.

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

A: Yes, you can simplify an expression with variables in the denominator. To do this, you can follow the same steps as before, but be careful not to divide by zero.

Q: How do I simplify an expression with negative exponents?

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

1. Rewrite the expression with positive exponents by moving the negative exponent to the other side of the fraction.
2. Simplify the resulting expression, if possible.

Q: Can I simplify an expression with absolute value?

A: Yes, you can simplify an expression with absolute value. To do this, you can follow the same steps as before, but be careful to handle the absolute value correctly.

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

A: To simplify an expression with multiple variables, you can follow the same steps as before, but be careful to handle the variables correctly.

Q: Can I simplify an expression with a variable in the numerator and denominator?

A: Yes, you can simplify an expression with a variable in the numerator and denominator. To do this, you can follow the same steps as before, but be careful to handle the variable correctly.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify expressions with fractions, mixed numbers, and variables. Remember to find a common denominator, rewrite fractions with a common denominator, and combine fractions. With practice, you will become proficient in simplifying algebraic expressions and be able to tackle even the most complex problems.

Tips and Tricks

• Always find a common denominator when adding or subtracting fractions.
• Rewrite fractions with a common denominator to make it easier to combine them.
• Simplify the numerator after combining fractions.
• Practice, practice, practice! The more you practice, the more comfortable you will become with simplifying algebraic expressions.

Common Mistakes to Avoid

• Failing to find a common denominator when adding or subtracting fractions.
• Not rewriting fractions with a common denominator.
• Not simplifying the numerator after combining fractions.
• Not practicing regularly to develop your skills.

Real-World Applications

Simplifying algebraic expressions has many real-world applications, including:

• Calculating interest rates and investments
• Determining the area and perimeter of shapes
• Solving problems in physics and engineering
• Creating models for population growth and other real-world phenomena

By mastering the art of simplifying algebraic expressions, you will be able to tackle a wide range of problems and make informed decisions in your personal and professional life.