Chapter 7.3, Problem 74HP

To determine

To find: To write two numbers in scientific notation with a product of $1.3\times {10}^{-3}$ then name two numbers in scientific notation with a quotient of $1.3\times {10}^{-3}$ .

Answer to Problem 74HP

The solution of expression is in calculation.

Explanation of Solution

Given information:

The expression is write two numbers in scientific notation with a product of $1.3\times {10}^{-3}$ then name two numbers in scientific notation with a quotient of $1.3\times {10}^{-3}$ .

Calculation:

Calculate the expression

Let the two $a\times {10}^n$ and $b\times {10}^m$ .

If the product is to be $1.3\times {10}^{-3}$ , then

  $\begin{array}{c}\left(a\times {10}^n\right)\times \left(b\times {10}^m\right)=1.3\times {10}^{-3}\\ ab\times {10}^{n+m}=1.3\times {10}^{-3}\\ ab\times {10}^{n+m}=13\times {10}^{-4}\end{array}$

Then we get, $ab=13$ and $1=a<10$ and $1=b<10$ .

Select integers for $n$ and $m$ such that $n+m=-4.$

Since $2.5\times 5.2=13$ and $5+\left(-9\right)=-4.$

Then let the numbers be $2.5\times {10}^5$ and $5.2\times {10}^9.$

  $\begin{array}{c}\left(2.5\times {10}^5\right)\left(5.2\times {10}^{-9}\right)=\left(2.5\times 5.2\right)\left({10}^5\times {10}^{-9}\right)\\ =13\times {10}^{5-9}\\ =13\times {10}^{-4}\\ =1.3\times {10}^{-3}\end{array}$

If the quotient is to be $1.3\times {10}^{-3}$ , then

  $\begin{array}{l}\frac{\left(a\times {10}^n\right)}{\left(b\times {10}^m\right)}=1.3\times {10}^{-3}\\ \frac{a}{b}\times {10}^{n-m}=1.3\times {10}^{-3}\end{array}$

Then we get, $\frac{a}{b}=1.3$ and $1=a<10$ and $1=b<10$ .

Select integers for $n$ and $m$ such that $n-m=-3.$

Since $\frac{2.6}{2}=1.3$ and $3-6=-3$ .

Let the numbers be $2.6\times {10}^3$ and $2\times {10}^6.$

Find the quotient:

  $\frac{2.6\times {10}^3}{2\times {10}^6}=\left(\frac{2.6}{2}\right)\left(\frac{{10}^3}{{10}^6}\right)=1.3\times {10}^{3-6}=1.3\times {10}^{-3}$

Therefore the solution of expression is above.