Question

Question #1c88f

Answer 1

`1.2 * 10^(-11)`

Explanation:

The thing to remember when multiplying numbers written in scientific notation is that you must multiply the mantissae and the exponents separately.

For a number written in scientific notation, you have

`color(white)(aa)color(blue)(m) xx 10^(color(purple)(n) color(white)(a)stackrel(color(white)(aaaaaa))(larr))color(white)(acolor(black)("the")acolor(purple)("exponent")aa)`
`color(white)(a/acolor(black)(uarr)aaaa)`
`color(white)(color(black)("the")acolor(blue)("mantissa")a)`

In your case, you have

`color(blue)(3.0) * 10^color(purple)(-14)" " and " " color(blue)(4.0) * 10^color(purple)(2)`

This means that when you multiply these two numbers, you have

`color(blue)(3.0) * 10^color(purple)(-14) * color(blue)(4.0) * 10^color(purple)(2)`

`= (color(blue)(3.0 * 4.0)) * (10^color(purple)(-14) * 10^color(purple)(2))`

`= color(blue)(12) * 10^color(purple)((-14 + 2))`

`= color(blue)(12) * 10^color(purple)(-12)`

More often than not, you will be dealing with normalized scientific notation, for which

`1 <= |color(blue)(m)| < 10`

To express the result of the multiplication in normalized scientific notation, divide the mantissa by `10` and multiply by `10` by adding `1` to the exponent.

`color(blue)(12) * 10^color(purple)(-12)`

`= color(blue)(12)/10 * 10^color(purple)(-12) * 10`

`= color(blue)(1.2) * 10^color(purple)(-11)`

Therefore, you can say that

`3.0 * 10^(-14) * 4.0 * 10^(2) = 1.2 * 10^(-11)`

The answer is rounded to two sig figs, the number of sig figs you have for the two numbers.