Divide. Write your answer in scientific notation.
$\frac{8 \times 10^9}{2 \times 10^3}$
We are tasked with dividing \(8 \times 10^9\) by \(2 \times 10^3\) and expressing the result in scientific notation. Scientific notation is written in the form \(a \times 10^n\), where \(1 \leq |a| < 10\) and \(n\) is an integer. The given expression is:
\[\frac{8 \times 10^9}{2 \times 10^3}\]
To simplify this, we can separate the constant coefficients and the powers of ten.
We can rewrite the expression by grouping the numbers and the powers of 10:
\[\frac{8 \times 10^9}{2 \times 10^3} = \left(\frac{8}{2}\right) \times \left(\frac{10^9}{10^3}\right)\]
First, divide the coefficients:
\[\frac{8}{2} = 4\]
Next, use the laws of exponents to divide the powers of ten. The rule \(\frac{x^a}{x^b} = x^{a-b}\) applies here:
\[\frac{10^9}{10^3} = 10^{9-3} = 10^6\]
Combining these two results, we get \(4 \times 10^6\). This is already in scientific notation, as \(1 \leq 4 < 10\).
The division of \(8 \times 10^9\) by \(2 \times 10^3\) yields \(4 \times 10^6\). The result is correctly formatted in scientific notation as \(4 \times 10^6\).
Final answer: \(\boxed{4 \times 10^6}\).
Scientific notation is used in science and engineering to handle very large or very small numbers conveniently. For example, if you are calculating the total distance light travels in a certain time, you might deal with numbers like the speed of light (\(3 \times 10^8\) meters per second) and time in seconds. Dividing or multiplying such numbers using scientific notation simplifies the arithmetic immensely, allowing us to quickly estimate and calculate values like the total distance in a compact form.