Find The Product. Simplify Your Answer Completely. ( M − 15 ) 2 M 2 − 30 M + 225 \frac{(m-15)^2}{m^2-30m+225} M 2 − 30 M + 225 ( M − 15 ) 2 ​

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Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. These types of fractions can be challenging to simplify, but with the right approach, they can be broken down into simpler fractions. In this article, we will focus on simplifying a specific complex fraction: (m−15)2m2−30m+225\frac{(m-15)^2}{m^2-30m+225}.

Breaking Down the Complex Fraction

To simplify the complex fraction, we need to start by factoring the numerator and the denominator. The numerator is a perfect square trinomial, which can be factored as follows:

(m−15)2=(m−15)(m−15)(m-15)^2 = (m-15)(m-15)

The denominator is a quadratic expression that can be factored as follows:

m2−30m+225=(m−15)(m−15)m^2-30m+225 = (m-15)(m-15)

Simplifying the Complex Fraction

Now that we have factored the numerator and the denominator, we can simplify the complex fraction by canceling out the common factors. In this case, both the numerator and the denominator have a factor of (m−15)(m-15).

(m−15)2m2−30m+225=(m−15)(m−15)(m−15)(m−15)\frac{(m-15)^2}{m^2-30m+225} = \frac{(m-15)(m-15)}{(m-15)(m-15)}

Canceling Out Common Factors

Since both the numerator and the denominator have a factor of (m−15)(m-15), we can cancel it out to simplify the fraction.

(m−15)(m−15)(m−15)(m−15)=m−151\frac{(m-15)(m-15)}{(m-15)(m-15)} = \frac{m-15}{1}

Final Simplification

The final simplified form of the complex fraction is m−151\frac{m-15}{1}. This can be further simplified to m−15m-15.

Conclusion

Simplifying complex fractions requires a step-by-step approach. By factoring the numerator and the denominator, we can identify common factors and cancel them out to simplify the fraction. In this article, we simplified the complex fraction (m−15)2m2−30m+225\frac{(m-15)^2}{m^2-30m+225} to its final form: m−15m-15.

Real-World Applications

Simplifying complex fractions has numerous real-world applications in various fields, including mathematics, science, and engineering. For example, in physics, complex fractions are used to describe the motion of objects and the behavior of electrical circuits. In engineering, complex fractions are used to design and optimize systems, such as bridges and buildings.

Tips and Tricks

When simplifying complex fractions, it's essential to follow these tips and tricks:

  • Factor the numerator and the denominator to identify common factors.
  • Cancel out common factors to simplify the fraction.
  • Use algebraic manipulations to simplify the fraction further.
  • Check the final simplified form to ensure it's correct.

Common Mistakes to Avoid

When simplifying complex fractions, it's essential to avoid these common mistakes:

  • Failing to factor the numerator and the denominator.
  • Not canceling out common factors.
  • Not checking the final simplified form for correctness.

Conclusion

Simplifying complex fractions is a crucial skill in mathematics and other fields. By following the step-by-step approach outlined in this article, you can simplify complex fractions and apply them to real-world problems. Remember to factor the numerator and the denominator, cancel out common factors, and check the final simplified form for correctness.

Final Thoughts

Simplifying complex fractions is a challenging task, but with practice and patience, you can master it. Remember to stay focused, follow the steps, and check your work to ensure accuracy. With this article, you now have the tools and knowledge to simplify complex fractions and tackle more complex problems in mathematics and other fields.

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Frequently Asked Questions

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to factor the numerator and the denominator, cancel out common factors, and check the final simplified form for correctness.

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

A: Some common mistakes to avoid when simplifying complex fractions include failing to factor the numerator and the denominator, not canceling out common factors, and not checking the final simplified form for correctness.

Q: Can you provide an example of a complex fraction?

A: Yes, here's an example of a complex fraction: (m−15)2m2−30m+225\frac{(m-15)^2}{m^2-30m+225}.

Q: How do I factor the numerator and the denominator of a complex fraction?

A: To factor the numerator and the denominator of a complex fraction, you need to identify the common factors and express them as a product of their greatest common factor (GCF) and a remaining expression.

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

A: The greatest common factor (GCF) of two expressions is the largest expression that divides both expressions without leaving a remainder.

Q: Can you provide an example of how to find the GCF of two expressions?

A: Yes, here's an example of how to find the GCF of two expressions: Let's say we have two expressions: 6x2+12x6x^2 + 12x and 3x2+6x3x^2 + 6x. To find the GCF, we need to identify the common factors and express them as a product of their GCF and a remaining expression. In this case, the GCF is 3x3x.

Q: How do I cancel out common factors in a complex fraction?

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

Q: What is the final simplified form of a complex fraction?

A: The final simplified form of a complex fraction is the expression that remains after canceling out common factors and simplifying the fraction.

Q: Can you provide an example of how to simplify a complex fraction?

A: Yes, here's an example of how to simplify a complex fraction: Let's say we have a complex fraction: (m−15)2m2−30m+225\frac{(m-15)^2}{m^2-30m+225}. To simplify this fraction, we need to factor the numerator and the denominator, cancel out common factors, and check the final simplified form for correctness. In this case, the final simplified form is m−15m-15.

Q: What are some real-world applications of simplifying complex fractions?

A: Some real-world applications of simplifying complex fractions include physics, engineering, and mathematics. In physics, complex fractions are used to describe the motion of objects and the behavior of electrical circuits. In engineering, complex fractions are used to design and optimize systems, such as bridges and buildings.

Q: Can you provide some tips and tricks for simplifying complex fractions?

A: Yes, here are some tips and tricks for simplifying complex fractions:

  • Factor the numerator and the denominator to identify common factors.
  • Cancel out common factors to simplify the fraction.
  • Use algebraic manipulations to simplify the fraction further.
  • Check the final simplified form for correctness.

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

A: Some common mistakes to avoid when simplifying complex fractions include failing to factor the numerator and the denominator, not canceling out common factors, and not checking the final simplified form for correctness.

Q: Can you provide some practice problems for simplifying complex fractions?

A: Yes, here are some practice problems for simplifying complex fractions:

  • Simplify the complex fraction: (x+5)2x2+10x+25\frac{(x+5)^2}{x^2+10x+25}
  • Simplify the complex fraction: (y−3)2y2−6y+9\frac{(y-3)^2}{y^2-6y+9}
  • Simplify the complex fraction: (z+2)2z2+4z+4\frac{(z+2)^2}{z^2+4z+4}

Q: How do I check the final simplified form of a complex fraction for correctness?

A: To check the final simplified form of a complex fraction for correctness, you need to verify that the simplified form is equivalent to the original complex fraction. You can do this by multiplying the numerator and the denominator of the simplified form by the common factor that was canceled out. If the result is equal to the original complex fraction, then the simplified form is correct.