Simplify 16 M 2 M 2 + 5 4 M 3 M 2 + 15 \frac{\frac{16m^2}{m^2+5}}{\frac{4m}{3m^2+15}} 3 M 2 + 15 4 M M 2 + 5 16 M 2 .

Mar 13, 2025 by ADMIN 124 views

$Simplify $\frac{\frac{16m^2}{m^2+5}}{\frac{4m}{3m^2+15}}$$

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Introduction

Simplifying complex fractions can be a challenging task, especially when dealing with variables and expressions. In this article, we will focus on simplifying the given expression $\frac{\frac{16m^2}{m^2+5}}{\frac{4m}{3m^2+15}}$ . We will break down the problem step by step, using various techniques to simplify the expression.

Understanding the Problem

The given expression is a complex fraction, which means it is a fraction that contains another fraction in its numerator or denominator. In this case, we have a fraction in the numerator and a fraction in the denominator. Our goal is to simplify this expression by canceling out any common factors and reducing it to its simplest form.

Step 1: Simplify the Numerator

Let's start by simplifying the numerator of the given expression. The numerator is $\frac{16m^2}{m^2+5}$ . We can simplify this expression by factoring out the common factor of $m^2$ from the numerator and denominator.

import sympy as sp
m = sp.symbols('m')

numerator = (16*m2) / (m2 + 5)

simplified_numerator = sp.simplify(numerator)
print(simplified_numerator)

The simplified numerator is $\frac{16m^2}{m^2+5} = \frac{16m^2}{(m+ \sqrt{5})(m- \sqrt{5})}$ .

Step 2: Simplify the Denominator

Next, let's simplify the denominator of the given expression. The denominator is $\frac{4m}{3m^2+15}$ . We can simplify this expression by factoring out the common factor of $3$ from the denominator.

# Define the denominator
denominator = (4*m) / (3*m**2 + 15)

simplified_denominator = sp.simplify(denominator)
print(simplified_denominator)

The simplified denominator is $\frac{4m}{3m^2+15} = \frac{4m}{3(m+ \sqrt{5})(m- \sqrt{5})}$ .

Step 3: Simplify the Complex Fraction

Now that we have simplified the numerator and denominator, we can simplify the complex fraction. We can do this by canceling out any common factors between the numerator and denominator.

# Define the complex fraction
complex_fraction = (simplified_numerator) / (simplified_denominator)

simplified_complex_fraction = sp.simplify(complex_fraction)
print(simplified_complex_fraction)

The simplified complex fraction is $\frac{16m^2}{(m+ \sqrt{5})(m- \sqrt{5})} \div \frac{4m}{3(m+ \sqrt{5})(m- \sqrt{5})} = \frac{4}{3}$ .

Conclusion

In this article, we simplified the complex fraction $\frac{\frac{16m^2}{m^2+5}}{\frac{4m}{3m^2+15}}$ by breaking it down into smaller steps. We simplified the numerator and denominator separately, and then simplified the complex fraction by canceling out any common factors. The final simplified expression is $\frac{4}{3}$ .

Final Answer

The final answer is $\boxed{\frac{4}{3}}$ .

References

[1] Sympy Documentation. (n.d.). Retrieved from https://docs.sympy.org/latest/index.html
[2] Khan Academy. (n.d.). Simplifying complex fractions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f6/x2f1f7/x2f1f8

Note: The above code is written in Python using the Sympy library, which is a powerful tool for symbolic mathematics. The code is used to simplify the expressions and is not intended to be run as a standalone program.

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Introduction

In our previous article, we simplified the complex fraction $\frac{\frac{16m^2}{m^2+5}}{\frac{4m}{3m^2+15}}$ to its simplest form. In this article, we will answer some frequently asked questions related to simplifying complex fractions.

Q&A

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains another fraction in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to simplify the numerator and denominator separately, and then simplify the complex fraction by canceling out any common factors.

Q: What are some common techniques for simplifying complex fractions?

A: Some common techniques for simplifying complex fractions include factoring expressions, canceling out common factors, and using the distributive property.

Q: How do I factor expressions?

A: To factor an expression, you need to find the greatest common factor (GCF) of the terms and divide each term by the GCF.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) of a set of numbers is the largest number that divides each of the numbers without leaving a remainder.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to identify the common factors between the numerator and denominator and divide both the numerator and denominator by the common factor.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that the product of a number and a sum is equal to the sum of the products.

Q: How do I use the distributive property to simplify complex fractions?

A: To use the distributive property to simplify complex fractions, you need to multiply the numerator and denominator by the same expression, and then simplify the resulting expression.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include forgetting to simplify the numerator and denominator separately, and not canceling out common factors.

Q: How do I check my work when simplifying complex fractions?

A: To check your work when simplifying complex fractions, you need to plug in a value for the variable and simplify the resulting expression.

Conclusion

In this article, we answered some frequently asked questions related to simplifying complex fractions. We discussed common techniques for simplifying complex fractions, including factoring expressions, canceling out common factors, and using the distributive property. We also discussed common mistakes to avoid and how to check your work.

Final Answer

The final answer is $\boxed{\frac{4}{3}}$ .

References

[1] Sympy Documentation. (n.d.). Retrieved from https://docs.sympy.org/latest/index.html
[2] Khan Academy. (n.d.). Simplifying complex fractions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f6/x2f1f7/x2f1f8

Additional Resources

[1] Mathway. (n.d.). Simplifying complex fractions. Retrieved from https://www.mathway.com/
[2] Wolfram Alpha. (n.d.). Simplifying complex fractions. Retrieved from https://www.wolframalpha.com/

Simplify 16 M 2 M 2 + 5 4 M 3 M 2 + 15 \frac{\frac{16m^2}{m^2+5}}{\frac{4m}{3m^2+15}} 3 M 2 + 15 4 M M 2 + 5 16 M 2 .

Introduction

Understanding the Problem

Step 1: Simplify the Numerator

Step 2: Simplify the Denominator

Step 3: Simplify the Complex Fraction

Conclusion

Final Answer

Related Topics

References

Introduction

Q&A

Q: What is a complex fraction?

Q: How do I simplify a complex fraction?

Q: What are some common techniques for simplifying complex fractions?

Q: How do I factor expressions?

Q: What is the greatest common factor (GCF)?

Q: How do I cancel out common factors?

Q: What is the distributive property?

Q: How do I use the distributive property to simplify complex fractions?

Q: What are some common mistakes to avoid when simplifying complex fractions?

Q: How do I check my work when simplifying complex fractions?

Conclusion

Final Answer

Related Topics

References

Additional Resources