Let's choose the scientific number \( 3.2 \times 10^4 \) and multiply it by \( 256.7 \).
Step 1: Set Up the Multiplication
We want to calculate: \[ (3.2 \times 10^4) \times 256.7 \]
Step 2: Multiply the Coefficients
First, multiply the coefficient \( 3.2 \) by \( 256.7 \):
\[ 3.2 \times 256.7 \]
Calculating this gives: \[ 3.2 \times 256.7 = 822.24 \]
Step 3: Combine with the Power of 10
Now we need to take the result and multiply it by \( 10^4 \): \[ 822.24 \times 10^4 \]
Step 4: Convert to Scientific Notation
Next, convert \( 822.24 \) into scientific notation.
Since \( 822.24 \) can be written as \( 8.2224 \times 10^2 \), we can combine this with \( 10^4 \):
\[ 822.24 \times 10^4 = (8.2224 \times 10^2) \times 10^4 \]
Step 5: Add the Exponents
Using the property of exponents \( a^m \times a^n = a^{m+n} \), we can combine the powers of ten:
\[ 8.2224 \times 10^{2+4} = 8.2224 \times 10^6 \]
Final Result
Thus, the final result of multiplying \( 3.2 \times 10^4 \) by \( 256.7 \) is:
\[ \boxed{8.2224 \times 10^6} \]