Question

Question #a04c1

Answer 1

`4.56 * 10^(-3)`

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 `1 = color(blue)(10)/color(purple)(10)` `->` keep in mind that we can multiply the number by `1` because that leaves its value 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 `color(darkorange)("(*)")`.

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(darkorange)("(*)")` is still not satisfied, so you must repeat the process again. This time, you have

`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 `3` sig figs, just like the number written in standard form.

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