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
#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.
