your answer is correct but not simlified yetgreg1313 said:\(\displaystyle A=\prod_{n=1}^{2015}\dfrac{n(n+3)+\sqrt2}{(n+1)^2-2}=\prod_{n=1}^{2015}\dfrac{n^2+3n+\sqrt2}{n^2+2n-1}\)
\(\displaystyle =\prod_{n=1}^{2015}\dfrac{(n-(-1-\sqrt2))(n-(-2+\sqrt2)}{(n-(-1-\sqrt2))(n-(-1+\sqrt2))}=\prod_{n=1}^{2015}\dfrac{n+2-\sqrt2}{n+1-\sqrt2}\)
\(\displaystyle =\dfrac{3-\sqrt2}{2-\sqrt2}\cdot\dfrac{4-\sqrt2}{3-\sqrt2}\cdot\dfrac{5-\sqrt2}{4-\sqrt2}\dots=\dfrac{2017-\sqrt2}{2-\sqrt2}\)
The process for simplifying and multiplying fractions involves the following steps:
Yes, fractions with different denominators can be multiplied and simplified. The resulting fraction will have a denominator that is the product of the original denominators.
It is not necessary to convert mixed numbers to improper fractions before multiplying and simplifying. However, it may be helpful to do so to make the simplification process easier.
If the resulting fraction cannot be simplified any further, then it is already in its simplest form. You can check this by making sure the numerator and denominator have no common factors other than 1.
No, there is no specific order in which the fractions should be multiplied. You can multiply them in any order, as long as you follow the steps for simplification afterwards.