# Simplify $A$: Multiply Fractions

• MHB
• Albert1
In summary, the process for simplifying and multiplying fractions involves multiplying the numerators and denominators, then simplifying the resulting fraction. Fractions with different denominators can be multiplied and simplified, and it is not necessary to convert mixed numbers to improper fractions. If the resulting fraction cannot be simplified further, it is already in its simplest form. There is no specific order in which the fractions should be multiplied.
Albert1
$A=(\dfrac {1\times 4+\sqrt 2}{2\times 2-2})\times (\dfrac {2\times 5+\sqrt 2}{3\times 3-2})\times(\dfrac {3\times 6+\sqrt 2}{4\times 4-2})\times --------\times (\dfrac {2015\times 2018+\sqrt 2}{{2016}\times {2016}-2})$

please simplify $A$

$$\displaystyle A=\prod_{n=1}^{2015}\dfrac{n(n+3)+\sqrt2}{(n+1)^2-2}=\prod_{n=1}^{2015}\dfrac{n^2+3n+\sqrt2}{n^2+2n-1}$$

$$\displaystyle =\prod_{n=1}^{2015}\dfrac{(n-(-1-\sqrt2))(n-(-2+\sqrt2)}{(n-(-1-\sqrt2))(n-(-1+\sqrt2))}=\prod_{n=1}^{2015}\dfrac{n+2-\sqrt2}{n+1-\sqrt2}$$

$$\displaystyle =\dfrac{3-\sqrt2}{2-\sqrt2}\cdot\dfrac{4-\sqrt2}{3-\sqrt2}\cdot\dfrac{5-\sqrt2}{4-\sqrt2}\dots=\dfrac{2017-\sqrt2}{2-\sqrt2}$$

greg1313 said:
$$\displaystyle A=\prod_{n=1}^{2015}\dfrac{n(n+3)+\sqrt2}{(n+1)^2-2}=\prod_{n=1}^{2015}\dfrac{n^2+3n+\sqrt2}{n^2+2n-1}$$

$$\displaystyle =\prod_{n=1}^{2015}\dfrac{(n-(-1-\sqrt2))(n-(-2+\sqrt2)}{(n-(-1-\sqrt2))(n-(-1+\sqrt2))}=\prod_{n=1}^{2015}\dfrac{n+2-\sqrt2}{n+1-\sqrt2}$$

$$\displaystyle =\dfrac{3-\sqrt2}{2-\sqrt2}\cdot\dfrac{4-\sqrt2}{3-\sqrt2}\cdot\dfrac{5-\sqrt2}{4-\sqrt2}\dots=\dfrac{2017-\sqrt2}{2-\sqrt2}$$
$A=P+Q\sqrt 2$
$P=? ,Q=?$

Last edited:
$$\displaystyle P=2016,\quad\,Q=\dfrac{2015}{2}$$

## What is the process for simplifying and multiplying fractions?

The process for simplifying and multiplying fractions involves the following steps:

• Multiply the numerators of the fractions.
• Multiply the denominators of the fractions.
• Simplify the resulting fraction by dividing the numerator and denominator by their greatest common factor (GCF).

## Can fractions with different denominators be multiplied and simplified?

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.

## Do I need to convert mixed numbers to improper fractions before multiplying and simplifying?

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.

## What should I do if the resulting fraction cannot be simplified any further?

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.

## Is there a specific order in which the fractions should be multiplied?

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.

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