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)aam×10naaaaaaa←−−−atheaexponentaa
color(white)(a/acolor(black)(uarr)aaaa)aa↑⏐ ⏐⏐aaaa
color(white)(color(black)("the")acolor(blue)("mantissa")a)theamantissaa
In your case, you have
color(blue)(3.0) * 10^color(purple)(-14)" " and " " color(blue)(4.0) * 10^color(purple)(2)3.0⋅10−14 and 4.0⋅102
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)3.0⋅10−14⋅4.0⋅102
= (color(blue)(3.0 * 4.0)) * (10^color(purple)(-14) * 10^color(purple)(2))=(3.0⋅4.0)⋅(10−14⋅102)
= color(blue)(12) * 10^color(purple)((-14 + 2))=12⋅10(−14+2)
= color(blue)(12) * 10^color(purple)(-12)=12⋅10−12
More often than not, you will be dealing with normalized scientific notation, for which
1 <= |color(blue)(m)| < 101≤|m|<10
To express the result of the multiplication in normalized scientific notation, divide the mantissa by
color(blue)(12) * 10^color(purple)(-12)12⋅10−12
= color(blue)(12)/10 * 10^color(purple)(-12) * 10=1210⋅10−12⋅10
= color(blue)(1.2) * 10^color(purple)(-11)=1.2⋅10−11
Therefore, you can say that
3.0 * 10^(-14) * 4.0 * 10^(2) = 1.2 * 10^(-11)3.0⋅10−14⋅4.0⋅102=1.2⋅10−11
The answer is rounded to two sig figs, the number of sig figs you have for the two numbers.
