Simplifying the Product of Two Fractions

Oct 13, 2025 by recoveryshake.com 140 views

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves combining like terms, canceling out common factors, and reducing fractions to their simplest form. In this article, we will focus on simplifying the product of two fractions, using the given expressions as examples.

Simplifying the Product of Two Fractions

To simplify the product of two fractions, we need to follow a few simple steps:

Multiply the numerators (the numbers on top) together.
Multiply the denominators (the numbers on the bottom) together.
Simplify the resulting fraction by canceling out any common factors.

Let's apply these steps to the given expressions.

Expression 1: $\frac{25x^2}{2x+6} \cdot \frac{2}{5x}$

To simplify this expression, we need to multiply the numerators and denominators together.

$\frac{25x^2}{2x+6} \cdot \frac{2}{5x} = \frac{25x^2 \cdot 2}{(2x+6) \cdot 5x}$

Now, let's simplify the numerator and denominator separately.

Numerator: $25x^2 \cdot 2 = 50x^2$

Denominator: $(2x+6) \cdot 5x = 10x^2 + 30x$

So, the simplified expression becomes:

$\frac{50x^2}{10x^2 + 30x}$

Expression 2: $\frac{25x^2}{2(x+3)} \cdot \frac{2}{5x}$

To simplify this expression, we need to multiply the numerators and denominators together.

$\frac{25x^2}{2(x+3)} \cdot \frac{2}{5x} = \frac{25x^2 \cdot 2}{2(x+3) \cdot 5x}$

Now, let's simplify the numerator and denominator separately.

Numerator: $25x^2 \cdot 2 = 50x^2$

Denominator: $2(x+3) \cdot 5x = 10x^2 + 30x$

So, the simplified expression becomes:

$\frac{50x^2}{10x^2 + 30x}$

Canceling Out Common Factors

Now that we have simplified both expressions, let's cancel out any common factors.

For Expression 1:

$\frac{50x^2}{10x^2 + 30x} = \frac{5x^2}{x^2 + 3x}$

We can cancel out the common factor of $x^2$ from the numerator and denominator.

$\frac{5x^2}{x^2 + 3x} = \frac{5}{x + 3}$

For Expression 2:

$\frac{50x^2}{10x^2 + 30x} = \frac{5x^2}{x^2 + 3x}$

We can cancel out the common factor of $x^2$ from the numerator and denominator.

$\frac{5x^2}{x^2 + 3x} = \frac{5}{x + 3}$

Conclusion

In conclusion, both expressions simplify to the same result: $\frac{5}{x + 3}$ . This is the simplified product of the two fractions.

Final Answer

The final answer is: $\boxed{\frac{5}{x + 3}}$

Discussion

The discussion category for this problem is mathematics, specifically algebra and calculus. The problem requires the application of algebraic techniques, such as simplifying fractions and canceling out common factors.

Q&A: Simplifying Algebraic Expressions

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

A: The first step in simplifying an algebraic expression is to combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to multiply the numerator and denominator by the same value. This will eliminate any common factors and reduce the fraction to its simplest form.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the number on top of a fraction, while the denominator is the number on the bottom. The numerator is multiplied by the denominator to get the final result.

Q: Can I cancel out any terms in an algebraic expression?

A: Yes, you can cancel out any terms that have a common factor. This involves dividing both the numerator and denominator by the common factor.

Q: How do I know when to cancel out terms in an algebraic expression?

A: You should cancel out terms in an algebraic expression when you see a common factor in both the numerator and denominator. This will simplify the expression and make it easier to work with.

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

A: The final step in simplifying an algebraic expression is to check if there are any remaining common factors. If there are, you should cancel them out to get the final simplified expression.

Common Mistakes to Avoid

1. Not Combining Like Terms

One common mistake to avoid is not combining like terms. This can lead to an incorrect simplified expression.

2. Not Canceling Out Common Factors

Another common mistake to avoid is not canceling out common factors. This can also lead to an incorrect simplified expression.

3. Not Checking for Remaining Common Factors

A third common mistake to avoid is not checking for remaining common factors. This can lead to an incorrect simplified expression.

Tips and Tricks

1. Use a Common Denominator

One tip to keep in mind is to use a common denominator when simplifying fractions. This will make it easier to cancel out common factors.

2. Factor Out Common Terms

Another tip to keep in mind is to factor out common terms. This will make it easier to cancel out common factors.

3. Check Your Work

A third tip to keep in mind is to check your work. This will ensure that you have simplified the expression correctly.

Conclusion

Simplifying algebraic expressions is an important skill in mathematics. By following the steps outlined in this article, you can simplify expressions and make them easier to work with. Remember to combine like terms, cancel out common factors, and check for remaining common factors. With practice, you will become more comfortable simplifying algebraic expressions and be able to tackle more complex problems.

Final Answer

The final answer is: $\boxed{\frac{5}{x + 3}}$