Simplify The Expression:$\[ \frac{7x}{3y} \cdot \frac{12y}{5x^2} + \frac{2}{3} \\]

Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying structure. In this article, we will explore the process of simplifying a given expression, using algebraic manipulation and mathematical reasoning. We will break down the expression into manageable parts, apply various algebraic rules, and ultimately arrive at a simplified form.

Understanding the Expression

The given expression is:

$$\frac{7x}{3y} \cdot \frac{12y}{5x^2} + \frac{2}{3}$$

This expression involves multiplication and addition of fractions, as well as variables and constants. To simplify this expression, we need to apply various algebraic rules and techniques.

Step 1: Multiply the Fractions

To simplify the expression, we start by multiplying the fractions together. When multiplying fractions, we multiply the numerators together and the denominators together.

$$\frac{7x}{3y} \cdot \frac{12y}{5x^2} = \frac{7x \cdot 12y}{3y \cdot 5x^2}$$

Step 2: Simplify the Numerator and Denominator

Now, we simplify the numerator and denominator separately. The numerator is $7x \cdot 12y$, which can be simplified to $84xy$. The denominator is $3y \cdot 5x^2$, which can be simplified to $15x^2y$.

$$\frac{7x \cdot 12y}{3y \cdot 5x^2} = \frac{84xy}{15x^2y}$$

Step 3: Cancel Out Common Factors

Now, we look for common factors between the numerator and denominator. In this case, we can cancel out the $x$ and $y$ terms.

$$\frac{84xy}{15x^2y} = \frac{84}{15x}$$

Step 4: Add the Remaining Term

Now, we add the remaining term, $\frac{2}{3}$, to the simplified fraction.

$$\frac{84}{15x} + \frac{2}{3}$$

Step 5: Find a Common Denominator

To add these fractions, we need to find a common denominator. The least common multiple of $15x$ and $3$ is $45x$. We can rewrite each fraction with this common denominator.

$$\frac{84}{15x} = \frac{84 \cdot 3}{15x \cdot 3} = \frac{252}{45x}$$

$$\frac{2}{3} = \frac{2 \cdot 15x}{3 \cdot 15x} = \frac{30x}{45x}$$

Step 6: Add the Fractions

Now, we can add the fractions together.

$$\frac{252}{45x} + \frac{30x}{45x} = \frac{252 + 30x}{45x}$$

Step 7: Simplify the Result

Finally, we simplify the result by combining like terms.

$$\frac{252 + 30x}{45x} = \frac{30x + 252}{45x}$$

Conclusion

In this article, we simplified a complex algebraic expression using various algebraic rules and techniques. We broke down the expression into manageable parts, applied various algebraic rules, and ultimately arrived at a simplified form. By following these steps, we can simplify any algebraic expression and reveal its underlying structure.

Final Answer

The final answer is $\boxed{\frac{30x + 252}{45x}}$.

Additional Tips and Tricks

• When simplifying algebraic expressions, always look for common factors between the numerator and denominator.
• Use the distributive property to expand expressions and simplify them.
• Apply the order of operations (PEMDAS) to simplify expressions.
• Use algebraic rules, such as the commutative and associative properties, to simplify expressions.

Real-World Applications

Algebraic expressions are used in a wide range of real-world applications, including:

• Physics and engineering: Algebraic expressions are used to describe the motion of objects and the behavior of physical systems.
• Economics: Algebraic expressions are used to model economic systems and make predictions about economic trends.
• Computer science: Algebraic expressions are used to describe algorithms and data structures.

Common Mistakes to Avoid

• When simplifying algebraic expressions, avoid canceling out terms that are not common factors.
• Avoid using the distributive property incorrectly, as this can lead to incorrect simplifications.
• Be careful when applying algebraic rules, as these can lead to incorrect simplifications if not used correctly.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics and has many real-world applications. By following the steps outlined in this article, we can simplify complex algebraic expressions and reveal their underlying structure. Remember to always look for common factors, use the distributive property, and apply algebraic rules correctly to simplify expressions.

Introduction

In our previous article, we explored the process of simplifying a complex algebraic expression using various algebraic rules and techniques. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

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

A: The first step in simplifying an algebraic expression is to look for common factors between the numerator and denominator. This can include variables, constants, or a combination of both.

Q: How do I simplify a fraction with variables in the numerator and denominator?

A: To simplify a fraction with variables in the numerator and denominator, you can cancel out common factors between the numerator and denominator. For example, if you have the fraction $\frac{xy}{x^2y}$, you can cancel out the $x$ and $y$ terms to get $\frac{1}{x}$.

Q: What is the distributive property, and how do I use it to simplify expressions?

A: The distributive property is a rule that states that a single term can be multiplied by each term in a product. For example, if you have the expression $3(x + y)$, you can use the distributive property to expand it to $3x + 3y$.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator. The least common multiple (LCM) of the denominators is the smallest number that both denominators can divide into evenly. Once you have found the common denominator, you can rewrite each fraction with that denominator and add them together.

Q: What is the order of operations, and how do I use it to simplify expressions?

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:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any 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 expression with multiple variables?

A: To simplify an expression with multiple variables, you can use the same techniques as you would for a single variable. Look for common factors between the numerator and denominator, use the distributive property, and apply algebraic rules as needed.

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

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

• Canceling out terms that are not common factors
• Using the distributive property incorrectly
• Applying algebraic rules incorrectly
• Forgetting to simplify expressions completely

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics and has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex algebraic expressions and reveal their underlying structure.

Additional Resources

• For more information on simplifying algebraic expressions, check out our previous article on the topic.
• For practice problems and exercises, try using online resources such as Khan Academy or Mathway.
• For more advanced topics in algebra, try checking out textbooks or online resources such as MIT OpenCourseWare.

Final Tips and Tricks

• Always look for common factors between the numerator and denominator.
• Use the distributive property to expand expressions and simplify them.
• Apply algebraic rules, such as the commutative and associative properties, to simplify expressions.
• Be careful when simplifying expressions with multiple variables.
• Practice, practice, practice! The more you practice simplifying algebraic expressions, the more comfortable you will become with the process.