Problem 91 - 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. $$\left(6.1 \times 10^{-8}\right)\left(2 \times 10^{-4}\right)$$

Step-by-Step Solution

Answer

The product of \(6.1 \times 10^{-8}\) and \(2 \times 10^{-4}\) is \(1.22 \times 10^{-11}\).

Step 1: Understanding the properties of scientific notation

A number in scientific notation is written as \(a \times 10^n\) where \(1 \leq |a| < 10\) and \(n\) is an integer. When multiplying two numbers in scientific notation, we multiply the coefficients and add the powers of 10.

Step 2: Multiplying the coefficients

Firstly, multiply the coefficients together. So, \( 6.1 \times 2 = 12.2\).

Step 3: Adding the exponents

Next, add the exponents of 10. So, \(-8 + -4 = -12\).

Step 4: Writing the answer in scientific notation

Combine the multiplied coefficients and added exponents to give the answer in scientific notation. However, \(12.2 \times 10^{-12}\) is not in proper scientific notation because \(12.2\) is greater than 10. To convert to proper scientific notation, divide the coefficient by 10 and add 1 to the exponent of 10. So, \(1.22 \times 10^{-11}\).