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Powers of ten can be useful while working with standard index numbers. Fill in the following table of powers of ten:

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Fill in a table of powers of 10.

When $n$ is positive, we multiply $10$ by itself $n$ times,

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\$\\text{e.g. } 10^3 = 10 \\times 10 \\times 10 = 1000 \\text{ .}\$

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When $n$ is negative, we can think of $10^{-n}$ as $\\frac{1}{10^{n}}$,

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\$\\text{e.g. } 10^{-3} = \\frac{1}{10^3} = \\frac{1}{1000} = 0.001\\text{ .}\$

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When $n = 0$:

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\$10^{0} = 1 \\text{ .}\$

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Generally, we can think of $10^n$ as a number in standard form $1 \\times 10^n$. Then $n$ always tells us the number of decimal places to move the decimal point in $1.0$, for example

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\$10^{-3} = 1.0 \\times 10^{-3} \\text{ and since } n = - 3 \\text{, we go } 3 \\text{ places back as follows: } 1.0 ⇒ 0.1 ⇒ 0.01 ⇒ 0.001 \\text{ .}\$

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A complete table of powers of ten for $n$ from $-6$ to $6$ is:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$n$$10^n -6$$0.000001$
$-5$$0.00001 -4$$0.0001$
$-3$$0.001 -2$$0.01$
$-1$$0.1 0$$1$
$1$$10 2$$100$
$3$$1000 4$$10000$
$5$$100000 6$$1000000$
\n

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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$n$$10^n [[0]]\\var{10^(-n-1)} \\var{-n+1}[[1]] 0$$1$
$1$$10$
$\\var{n}$[[2]]
[[3]]$\\var{10^(n+2)}$
\n

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