Chapter 9, Problem 46SGR

To determine

To calculate: To find how many nanometers are in $1$ millimeter and write using positive exponent

Answer to Problem 46SGR

There are $\frac{1}{{10}^6}$ nanometers in $1$ millimeter

Explanation of Solution

Given information: If $1$ millimeter is equal to ${10}^{-3}$ meter and $1$ nanometer is equal to ${10}^{-9}$ meter

Formula Used:

Exponential identities are

  $x^a{x}^b=x^{a+b}$

  $x^a{y}^a={\left(xy\right)}^a$

  ${\left(x^a\right)}^b={\left(x\right)}^{ab}$

  $\frac{x^a}{x^b}={\left(x\right)}^{a-b}$

  $x^{-a}=\frac{1}{x^a}$

Calculation:

Given that $1$ millimeter is equal to ${10}^{-3}$ meter and $1$ nanometer is equal to ${10}^{-9}$ meter

Expression to find number of nanometers in $1$ millimeter

  $\frac{{10}^{-9}}{{10}^{-3}}$

On solving the above expression using the exponential identity $\frac{x^a}{x^b}={\left(x\right)}^{a-b}$

  $\begin{array}{l}\frac{{10}^{-9}}{{10}^{-3}}={10}^{-9-\left(-3\right)}\\ \frac{{10}^{-9}}{{10}^{-3}}={10}^{-9+3}\\ \frac{{10}^{-9}}{{10}^{-3}}={10}^{-6}\end{array}$

Using the exponential identity $x^{-a}=\frac{1}{x^a}$ to write the expression with positive exponent

  ${10}^{-6}=\frac{1}{{10}^6}$

Conclusion:

Hence, thereare $\frac{1}{{10}^6}$ nanometers in $1$ millimeter