Problem 98 - College Algebra Essentials

Question

Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$\frac{1.2 \times 10^{4}}{2 \times 10^{-2}}$$

Step-by-Step Solution

Answer

The answer is \(6 \times 10^{5}\).

Step 1: Simplify the Numerator and Denominator

In the given problem, the numerator is \(1.2 \times 10^{4}\) and the denominator is \(2 \times 10^{-2}\). We can rewrite the division as a multiplication by applying the rule \(a \div b = a \times \frac{1}{b}\). So, the problem becomes: \(1.2 \times 10^{4} \times \frac{1}{2 \times 10^{-2}}\).

Step 2: Rewrite the Reciprocal of the Denominator

The reciprocal of the denominator \(\frac{1}{2 \times 10^{-2}}\) can be rewritten as \(\frac{1}{2} \times \frac{1}{10^{-2}}\). Further, this simplifies to \(0.5 \times 10^{2}\) since the exponent '-2' becomes positive when we take the reciprocal of it.

Step 3: Multiply through the numbers

First, multiply the numbers out in front, which is \(1.2 \times 0.5 = 0.6\). Then, when you multiply the powers of 10 together, you add the exponents. So, \(10^{4} \times 10^{2} = 10^{4+2} = 10^{6}\).

Step 4: Write the Final Answer in Scientific Notation

The product of the numbers is \(0.6 \times 10^{6}\). However, for scientific notation, the decimal place should be after the first digit. So, we adjust it to \(6 \times 10^{-1} \times 10^{6} = 6 \times 10^{6-1} = 6 \times 10^{5}\). We don't need to round the decimal factor in the answer because it's already at the desired precision.