1. ## approximation

Write down the first three terms in the binomial expansion of $\displaystyle \sqrt[4]{1+\frac{x}{81}}$
Here i got

$\displaystyle 1+\frac{x}{324}-\frac{x^2}{69984}$

Then deduce that $\displaystyle \sqrt[4]{80}=2.9907$ .. not sure with this part

That should be the approximation sign , i am not sure how to do it with latex .

2. Originally Posted by thereddevils
Write down the first three terms in the binomial expansion of $\displaystyle \sqrt[4]{1+\frac{x}{81}}$
Here i got

$\displaystyle 1+\frac{x}{324}-\frac{x^2}{69984}$

Then deduce that $\displaystyle \sqrt[4]{80}=2.9907$ .. not sure with this part

That should be the approximation sign , i am not sure how to do it with latex .
Note that $\displaystyle \sqrt[4]{1+\frac{x}{81}} = \sqrt[4]{\frac{81+x}{81}} = \frac{1}{3} \sqrt[4]{81+x}$ so substitute $\displaystyle x = -1$ into the approximate expansion of $\displaystyle \sqrt[4]{1+\frac{x}{81}}$ and then re-arrange to get the approximate value of $\displaystyle \sqrt[4]{80}$.

3. Originally Posted by mr fantastic
Note that $\displaystyle \sqrt[4]{1+\frac{x}{81}} = \sqrt[4]{\frac{81+x}{81}} = \frac{1}{3} \sqrt[4]{81+x}$ so substitute $\displaystyle x = -1$ into the approximate expansion of $\displaystyle \sqrt[4]{1+\frac{x}{81}}$ and then re-arrange to get the approximate value of $\displaystyle \sqrt[4]{80}$.

THank you .