# Question #a04c1

##### 1 Answer

#### Answer:

#### Explanation:

Your goal here is to write your number in *normalized scientific notation*, which has

#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)#

and

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

As you know, you have

#10^0 = 1#

This means that you can write your initial number as

#color(blue)(0.00456) * 10^color(purple)(0)#

Now, to start converting the number to scientific notation, multiply it by **unchanged**!

#color(blue)(0.00456) * 10^color(purple)(0) * color(blue)(10)/color(purple)(10)#

You can rewrite this as

#color(blue)(0.00456) * color(blue)(10) * 10^color(purple)(0)/color(purple)(10) = color(blue)(0.0456) * 10^color(purple)(-1)#

At this point, you must check to see if the new value of the mantissa satisfies condition

Since

#1 color(red)(cancel(color(black)(<=))) color(blue)(0.0456) color(red)(cancel(color(black)(<))) 10#

you must repeat the process again. This time, you have

#color(blue)(0.0456) * 10^color(purple)(-1) * color(blue)(10)/color(purple)(10)#

which is equivalent to

#color(blue)(0.0456) * color(blue)(10) * 10^color(purple)(-1)/color(purple)(10) = color(blue)(0.456) * 10^color(purple)(-2)#

Condition

#color(blue)(0.456) * 10^color(purple)(-2) * color(blue)(10)/color(purple)(10)#

which is equivalent to

#color(blue)(0.456) * color(blue)(10) * 10^color(purple)(-2)/color(purple)(10) = color(blue)(4.56) * 10^color(purple)(-3)#

Finally, you have

#1 <= color(blue)(4.56) < 10" " " "color(darkgreen)(sqrt())#

so you can say that

#color(darkgreen)(ul(color(black)(0.00456 = 4.56 * 10^(-3))))#

Notice that the number written in scientific notation has **sig figs**, just like the number written in standard form.

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