To find out how many feathers are needed to equal a weight of \( 4.1 \times 10^3 \) g, you can use the formula:
\[ \text{Number of feathers} = \frac{\text{Total weight}}{\text{Weight of one feather}} \]
Given:
- Total weight = \( 4.1 \times 10^3 \) g
- Weight of one feather = \( 0.0082 \) g
First, convert \( 0.0082 \) g into scientific notation:
\[ 0.0082 , \text{g} = 8.2 \times 10^{-3} , \text{g} \]
Now, substitute into the formula:
\[ \text{Number of feathers} = \frac{4.1 \times 10^3 , \text{g}}{8.2 \times 10^{-3} , \text{g}} \]
To divide the numbers in scientific notation, divide the coefficients and subtract the exponents:
\[ \text{Number of feathers} = \frac{4.1}{8.2} \times 10^{3 - (-3)} \]
Calculating the coefficients:
\[ \frac{4.1}{8.2} = 0.5 \]
Now, calculate the exponent:
\[ 3 - (-3) = 3 + 3 = 6 \]
So:
\[ \text{Number of feathers} = 0.5 \times 10^{6} \]
To convert \( 0.5 \) into scientific notation:
\[ 0.5 = 5.0 \times 10^{-1} \]
Thus:
\[ \text{Number of feathers} = 5.0 \times 10^{-1} \times 10^{6} = 5.0 \times 10^{5} \]
Therefore, the number of feathers needed is:
\[ \boxed{5.0 \times 10^{5}} \]