Chapter 7.2, Problem 42P

Simplify each expression. Write each answer in scientific notation.

$\left(0.2\times {10}^{11}\right)\left(0.4\times {10}^{-14}\right)$

To determine

To calculate: The simplified form of the expression $\left(0.2\times {10}^{11}\right)\left(0.4\times {10}^{-14}\right)$ and express it in scientific notation.

Answer to Problem 42P

The simplified form of the expression $\left(0.2\times {10}^{11}\right)\left(0.4\times {10}^{-14}\right)$ is $8\times {10}^{-5}$ .

Explanation of Solution

Given information:

The expression $\left(0.2\times {10}^{11}\right)\left(0.4\times {10}^{-14}\right)$ .

Formula used:

The scientific notation is of the form $\text{a}\text{.bcd}\times {\text{10}}^n$ where $\text{a}\text{.bcd}$ is a number between 1 and 10 which is multiplied by a power of 10.

Calculation:

Consider the expression $\left(0.2\times {10}^{11}\right)\left(0.4\times {10}^{-14}\right)$ .

The law of exponent states $a^m{a}^n=a^{m+n}$ . Apply it,

  $\begin{array}{c}\left(0.2\times {10}^{11}\right)\left(0.4\times {10}^{-14}\right)=0.2\times 0.4\times {10}^{11-14}\\ =0.08\times {10}^{-3}\end{array}$

Recall that scientific notation is of the form $\text{a}\text{.bcd}\times {\text{10}}^n$ where $\text{a}\text{.bcd}$ is a number that lie between 1 and 10 which is multiplied by a power of 10.

Therefore, in scientific notation it is expressed as,

  $\begin{array}{c}\left(0.2\times {10}^{11}\right)\left(0.4\times {10}^{-14}\right)=0.2\times 0.4\times {10}^{11-14}\\ =0.08\times {10}^{-3}\\ =0.08\times {10}^2\times {10}^{-2}\times {10}^{-3}\\ =8\times {10}^{-5}\end{array}$

Thus, the simplified form of the expression $\left(0.2\times {10}^{11}\right)\left(0.4\times {10}^{-14}\right)$ is $8\times {10}^{-5}$ .