Question #a04c1

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Question #a04c1

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Answer 1 · 2017-09-04T00:23:24

$4.56 \cdot 10^{- 3}$ $4.56 * 10^(-3)$

Explanation:

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

$a a m \times 10^{n a a a a a a a \leftarrow -- -} a the a exponent a a$ $color(white)(aa)color(blue)(m) xx 10^(color(purple)(n) color(white)(a)stackrel(color(white)(aaaaaa))(larr))color(white)(acolor(black)("the")acolor(purple)("exponent")aa)$
$\frac{a}{a} ↑ ⏐ ⏐ ⏐ a a a a$ $color(white)(a/acolor(black)(uarr)aaaa)$
$the a mantissa a$ $color(white)(color(black)("the")acolor(blue)("mantissa")a)$

and

$1 \leq | m | < 10 (*)$ $1 <= |color(blue)(m)| < 10" " " "color(darkorange)("(*)")$

As you know, you have

$10^{0} = 1$ $10^0 = 1$

This means that you can write your initial number as

$0.00456 \cdot 10^{0}$ $color(blue)(0.00456) * 10^color(purple)(0)$

Now, to start converting the number to scientific notation, multiply it by $1 = \frac{10}{10}$ $1 = color(blue)(10)/color(purple)(10)$ $\to$ $->$ keep in mind that we can multiply the number by $1$ $1$ because that leaves its value unchanged!

$0.00456 \cdot 10^{0} \cdot \frac{10}{10}$ $color(blue)(0.00456) * 10^color(purple)(0) * color(blue)(10)/color(purple)(10)$

You can rewrite this as

$0.00456 \cdot 10 \cdot \frac{10^{0}}{10} = 0.0456 \cdot 10^{- 1}$ $color(blue)(0.00456) * color(blue)(10) * 10^color(purple)(0)/color(purple)(10) = color(blue)(0.0456) * 10^color(purple)(-1)$

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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Question #a04c1

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