Multiply 0.75 by $2.8 \times 10^{-5}$
The problem asks us to compute the product of two numbers, \(0.75\) and \(2.8 \times 10^{-5}\), and express the result in scientific notation. Scientific notation is a standard way to write numbers as \(A \times 10^n\), where \(1 \le A < 10\) and \(n\) is an integer. Our target is to express the result clearly and identify which of the provided options matches our finding.
First, convert the decimal \(0.75\) into scientific notation. Since \(0.75\) is equivalent to \(7.5 \times 0.1\), we can write it as:
\[0.75 = 7.5 \times 10^{-1}.\]
Now, we set up the multiplication with the second number:
\[(7.5 \times 10^{-1}) \times (2.8 \times 10^{-5}).\]
By the associative and commutative properties of multiplication, we group the numerical coefficients and the powers of \(10\) separately:
\[(7.5 \times 2.8) \times (10^{-1} \times 10^{-5}).\]
Calculating the product of the coefficients:
\[7.5 \times 2.8 = 21.0.\]
Calculating the product of the powers of \(10\) using the exponent law \(10^a \times 10^b = 10^{a+b}\):
\[10^{-1} \times 10^{-5} = 10^{-1 + (-5)} = 10^{-6}.\]
Combining these, we get:
\[21.0 \times 10^{-6}.\]
To express \(21.0 \times 10^{-6}\) in standard scientific notation, we need the coefficient to be between 1 and 10. We can rewrite \(21.0\) as \(2.1 \times 10^1\). Substituting this back into our expression:
\[(2.1 \times 10^1) \times 10^{-6} = 2.1 \times 10^{1 - 6} = 2.1 \times 10^{-5}.\]
Comparing this result with the given choices, we find it matches the first option.
\[\boxed{2.1 \times 10^{-5}}\]
Scientific notation is widely used in fields like physics and chemistry to handle very large or very small numbers efficiently. For example, if a chemist is measuring the mass of a single molecule in grams, they might use this notation to keep track of extremely small quantities without dealing with long strings of zeros. This process of multiplying scientific expressions, as shown here, is fundamental to performing precise calculations in these scientific disciplines.