If $a$ and $b$ are non-zero integers, $m$ and $n$ are any integers, then which of the following are true?

\frac{a^{m}}{a^{n}} = a^{\frac{m}{n}}

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a^{m} \times a^{n} = a^{(m + n)}

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a^{m} \times b^{m} = {(a b)}^{m}

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\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}

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Solution

The correct options are
B $a^{m} \times a^{n} = a^{(m + n)}$
C $a^{m} \times b^{m} = {(a b)}^{m}$
D $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$
If $a$ and $b$ are non-zero integers, $m$ and $n$ are any integers, then

$\frac{a^{m}}{a^{n}} = a^{m - n},$

$a^{m} \times a^{n} = a^{m + n},$

$a^{m} \times b^{m} = {(a b)}^{m}$ and

$\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m} .$

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If $m$ and $n$ are integers, $a$ is a non-zero integer and $a^{m}$ × $a^{n}$ = $a^{x}$ , then the value of $x$ is _____.

\frac{{(a + \frac{1}{b})}^{m} \times {(a - \frac{1}{b})}^{n}}{{(b + \frac{1}{a})}^{m} \times {(b - \frac{1}{a})}^{n}} = {(\frac{a}{b})}^{m + n}

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