To find out how many feathers are needed to make a weight of \(4.1 \times 10^3\) grams, you can divide the total weight by the weight of one feather.
The weight of one feather is \(0.0082\) grams.
Now, perform the calculation:
\[ \text{Number of feathers} = \frac{4.1 \times 10^3 \text{ g}}{0.0082 \text{ g}} \]
Calculating the division, we start by rewriting \(0.0082\) in scientific notation:
\[ 0.0082 = 8.2 \times 10^{-3} \]
Now, substituting back into the equation:
\[ \text{Number of feathers} = \frac{4.1 \times 10^3}{8.2 \times 10^{-3}} \]
To simplify, divide the coefficients and subtract the exponents of the powers of ten:
\[ = \left(\frac{4.1}{8.2}\right) \times 10^{3 - (-3)} \]
Calculating \(\frac{4.1}{8.2}\):
\[ \frac{4.1}{8.2} = 0.5 \]
And for the powers of ten:
\[ 3 - (-3) = 3 + 3 = 6 \]
Putting it all together:
\[ \text{Number of feathers} = 0.5 \times 10^6 \]
Now, convert \(0.5\) to scientific notation:
\[ 0.5 = 5.0 \times 10^{-1} \]
Thus, multiplying by \(10^6\):
\[ 0.5 \times 10^6 = 5.0 \times 10^{-1} \times 10^6 = 5.0 \times 10^{5} \]
Therefore, the number of feathers needed is:
\[ \boxed{5.0 \times 10^{5}} \]