# How should 0.0000497 be expressed in proper scientific notation?

##### 1 Answer

#### Explanation:

For a number written using *normalized 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)#

For *normalized scientific notation*, you must have

#1 <= |m| < 10" "color(orange)(("*"))#

Your goal here is to find the value of the mantissa and the value of the exponent. To find the mantissa, start to move the decimal point **to the right** of the given number until you find a value that matches the given condition

For **every position** you move the decimal point * to the right*, you must subtract

You will have

#0.0000497 -> color(purple)(n)=color(white)(-)0#

#color(white)(0)0.000497 -> color(purple)(n)=-1" and " 1 color(red)(cancel(color(black)(<=)))color(blue)(m)=0.000497 < 10#

#color(white)(00)0.00497 -> color(purple)(n)=-2" and " 1 color(red)(cancel(color(black)(<=)))color(blue)(m)=color(white)(0)0.00497 < 10#

#color(white)(000)0.0497 -> color(purple)(n)=-3" and " 1 color(red)(cancel(color(black)(<=)))color(blue)(m)=color(white)(00)0.0497 < 10#

#color(white)(0000)0.497 -> color(purple)(n)=-4" and " 1 color(red)(cancel(color(black)(<=)))color(blue)(m)=color(white)(000)0.497 < 10#

#color(white)(00000)4.97 -> color(purple)(n)=-5" and " 1 <= color(white)(a)color(blue)(m)=color(white)(0000)4.97 < 10 " "color(darkgreen)(sqrt())#

And there you have it. You moved the decimal **five positions** to the * right*, which means that the exponent is equal to n =

#0.0000497 = color(green)(|bar(ul(color(white)(a/a)color(black)(4.97 * 10^(-5))color(white)(a/a)|)))#