# Question #1c88f

##### 1 Answer

#### 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 **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.