# How can you memorize exponent rules?

Mar 30, 2016

See explanation...

#### Explanation:

${a}^{n} = {\overbrace{a \times a \times \ldots \times a}}^{\text{n times}}$

Then you can see:

${a}^{m} \times {a}^{n} = {\overbrace{a \times a \times \ldots \times a}}^{\text{m times" xx overbrace(a xx a xx ... xx a)^"n times}}$

$= {\overbrace{a \times a \times \ldots \times a}}^{\text{m + n times}} = {a}^{m + n}$

This is useful when you multiply two numbers that are expressed in scientific notation. For example:

$\left(1.2 \times {10}^{3}\right) \times \left(2.4 \times {10}^{6}\right)$

$= \left(1.2 \times 2.4\right) \times \left({10}^{3} \times {10}^{6}\right)$

$= 2.88 \times {10}^{3 + 6}$

$= 2.88 \times {10}^{9}$

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For negative exponents, first note that if $a \ne 0$:

${a}^{- n} = \frac{1}{\underbrace{a \times a \times \ldots \times a}} _ \text{n times}$

and we find:

${a}^{n} \times {a}^{- n} = {\overbrace{a \times a \times \ldots \times a}}^{\text{n times" xx 1/underbrace(a xx a xx ... xx a)_"n times}}$

$= {\overbrace{a \times a \times \ldots \times a}}^{\text{n times"/underbrace(a xx a xx ... xx a)_"n times}} = 1$

We find that the rule: ${a}^{m} \times {a}^{n} = {a}^{m + n}$ works for any integer values of $m$ and $n$, positive, negative or $0$.

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The next level of complexity is:

${\left({a}^{m}\right)}^{n} = {\overbrace{{a}^{m} \times {a}^{m} \times . . \times {a}^{m}}}^{\text{n times}} = {a}^{m n}$

For example:

${\left({2}^{2}\right)}^{3} = {4}^{3} = 64$

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Finally note that ${a}^{{m}^{n}}$ is evaluated from right to left.

That is:

${a}^{{m}^{n}} = {a}^{\left({m}^{n}\right)}$

For example:

${2}^{{2}^{3}} = {2}^{8} = 256$