Let's simplify the expression \(2.7 \times 10^{5} \times 3.2 \times 10^{7}\) step by step.
Step 1: Multiply the coefficients.
The coefficients are \(2.7\) and \(3.2\).
\[ 2.7 \times 3.2 = 8.64 \]
Step 2: Simplify the exponents.
The exponents are \(10^{5}\) and \(10^{7}\). When multiplying with the same base, we add the exponents.
\[ 10^{5} \times 10^{7} = 10^{(5 + 7)} = 10^{12} \]
Step 3: Change into Scientific Notation.
Now we can combine the results from Steps 1 and 2:
\[ 8.64 \times 10^{12} \]
However, the number \(8.64\) does not fit the standard form of scientific notation, which requires the coefficient to be between \(1\) and \(10\). We can convert it:
\[ 8.64 = 8.64 \times 10^{0} = 0.864 \times 10^{1} \]
Now we adjust the notation accordingly:
\[ (0.864 \times 10^{1}) \times 10^{12} = 0.864 \times 10^{13} \]
Step 4: Final Answer
The simplified expression in scientific notation is:
\[ \boxed{8.64 \times 10^{12}} \]
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