Chapter P.1, Problem 64E

In Exercises 63 and 64, use scientific notation to simplify.

64. $\frac{\left(3.7\times {10}^{-7}\right)\left(4.3\times {10}^6\right)}{2.5\times {10}^7}$

Solution

To simplify the expression $\frac{\left(3.7\times {10}^{-7}\right)\left(4.3\times {10}^6\right)}{2.5\times {10}^7}$ using scientific notation.

  $6.364\times {10}^{-8}$

Given:

The expression $\frac{\left(3.7\times {10}^{-7}\right)\left(4.3\times {10}^6\right)}{2.5\times {10}^7}$ .

Concept Used:

• To write a number in scientific notation, move the decimal point after the first nonzero integer and raise the exponent of 10 by the number of digits we moved the decimal point. If the decimal point was moved to the left, then raise the positive exponent and if the decimal point moved to the right, then raise the negative exponent.
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of powers property: $\frac{a^m}{a^n}=a^{m-n}$

Calculation:

In order to simplify the given expression using scientific notation, use the exponent properties and simplify further as shown below

  $\begin{array}{c}\frac{\left(3.7\times {10}^{-7}\right)\left(4.3\times {10}^6\right)}{2.5\times {10}^7}=\frac{\left(3.7\times 4.3\right)\times \left({10}^{-7}\times {10}^6\right)}{2.5\times {10}^7}\\ =\frac{\left(15.91\right)\times \left({10}^{-7+6}\right)}{2.5\times {10}^7}\\ =\frac{15.91\times {10}^{-1}}{2.5\times {10}^7}\\ =\frac{15.91}{2.5}\times \frac{{10}^{-1}}{{10}^7}\\ =6.364\times {10}^{-1-7}\\ =6.364\times {10}^{-8}\end{array}$

Thus, the simplified expression is $6.364\times {10}^{-8}$ .