Problem 100 - Algebra and Trigonometry

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. $$\frac{7.5 \times 10^{-2}}{2.5 \times 10^{6}}$$

Step-by-Step Solution

Answer

The short answer to the problem is \(3 \times 10^{-2}\).

Step 1: Perform the Division

Divide the division part separately for the numerical part and the power part. So, we take \(7.5\) divided by \(2.5\) to get \(3\) for the numerical part of the answer. Then, we subtract the exponent in the denominator from the exponent in the numerator to get the power part. Therefore, we write \(-6\) subtracted from \(-2\) which gives \(-4\) for the power part.

Step 2: Get the result in scientific notation

Combine the numerical part and the power part to represent the scientific notation. Here, the numerical part is \(3\) and the power part is \(-4\), hence it results to \(3 \times 10^{-4}\).

Step 3: Multiply by 100

Next, we multiply the result by \(100\). So, we have \(100 \times 3 \times 10^{-4}\).

Step 4: Simplify

Multiplying \(100\) and \(3\) gives \(300\), so the expression becomes \(300 \times 10^{-4}\).

Step 5: Adjust to correct scientific notation

The resulting notation \(300 \times 10^{-4}\) should be adjusted to correct scientific notation which allows only one digit before the decimal. Divide the number \(300\) by \(10\), and at the same time increase the power of \(10\) by \(1\). Therefore, it becomes \(30 \times 10^{-3}\) and continuing this process, \(3 \times 10^{-2}\).