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The product of two powers with the same non-zero base $a$ and integer exponents $m$ and $n$ can be written as a single power with base $a$ and exponent $m+n.$

$a_{m}⋅a_{n}=a_{m+n}$

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, $a_{m}$ and $a_{n}$ will be written as products using the definition of a power.

$a_{m}a_{n} =mtimesa⋅a⋅…⋅a =ntimesa⋅a⋅…⋅a $

With this information, the left-hand side of the identity can be rewritten using multiplication.
$a_{m}⋅a_{n}=m+ntimesmtimesa⋅a⋅…⋅a ⋅ntimesa⋅a⋅…⋅a $

By the definition of a power, multiplying a number by itself $m+n$ times is equivalent to raising that number to the power of $m+n.$
$m+ntimesmtimesa⋅a⋅…⋅a ⋅ntimesa⋅a⋅…⋅a =a_{m+n} $

The Product of Powers Property has been proved. $a_{m}⋅a_{n}=a_{m+n}$

The ratio of two powers with the same non-zero base $a$ and integer exponents $m$ and $n$ can be written as a single power with base $a$ and exponent $m−n.$

$a_{n}a_{m} =a_{m−n}$

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, $a_{m}$ and $a_{n}$ will be written as products using the definition of a power.

$a_{n}a_{m} =ntimesa⋅a⋅a⋅…⋅a a⋅a⋅a⋅…⋅amtimes $

Next, the common factors will be eliminated. It will be arbitrarily assumed that $m>n.$
$a_{n}a_{m} =ntimesa ⋅a ⋅a ⋅…⋅a a ⋅a ⋅a ⋅…⋅amtimes $

Note that when the quotient is simplified, the number of remaining factors will be $m−n.$ $a_{n}a_{m} =m−ntimesa⋅a⋅…⋅a $

Finally, the definition of a power will be used one more time to express the right-hand side of the above equation as a single power. With this last step, the Quotient of Powers Property is proven. $a_{n}a_{m} =a_{m−n}$

Note that this property is also valid for $m≤n.$ If $m=n,$ then the exponent will be zero. If $m<n,$ the exponent will be negative.

A power with a non-zero base $a$ and an integer exponent $m$ that is raised to another integer exponent $n$ can be written as a power with base $a$ and exponent $m⋅n.$

$(a_{m})_{n}=a_{m⋅n}$

For the rule to be true for $a=0,$ both exponents must be greater than zero.

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, $(a_{m})_{n}$ will be rewritten as a product using the definition of a power.

$(a_{m})_{n}=ntimesa_{m}⋅a_{m}⋅⋯⋅a_{m} $

Then, all occurrences of the expression $a_{m}$ will be written as products using the definition of a power one more time.
$(a_{m})_{n}=mtimesa⋅⋯⋅a ⋅mtimesa⋅⋯⋅a ⋅⋯⋅mtimesa⋅⋯⋅a ↘↓↙ntimes $

Notice that the base $a$ is multiplied by itself $m⋅n$ times. By the definition of a power, the right-hand side of the above equation is equivalent to raising $a$ to the power of $m⋅n.$
$m⋅ntimesa⋅a⋅a⋅⋯⋅a =a_{m⋅n} $

The Power of a Power Property has been proved. $(a_{m})_{n}=a_{m⋅n}$

A power with an integer exponent $m$ whose base is the product of two non-zero factors $a$ and $b$ can be written as the product of two powers with bases $a$ and $b$ and the same exponent $m.$

$(ab)_{m}=a_{m}b_{m}$

For this rule to be valid when either $a$ or $b$ is $0,$ $m$ must be greater than zero.

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, $(ab)_{m}$ will be written as a product using the definition of a power.

$(ab)_{m}=mtimes(ab)⋅(ab)⋅⋯⋅(ab) $

Next, the parentheses can be removed.
$mtimes↗↑↖(ab)_{m}=a⋅b⋅a⋅b⋅⋯⋅a⋅b↘↓↙mtimes $

Now, the Commutative Property of Multiplication can be used.
$(ab)_{m}=mtimesa⋅a⋅⋯⋅a ⋅mtimesb⋅b⋅⋯⋅b $

Finally, using the definition of a power again, the right-hand side of the equation can be written as the product of $a_{m}$ and $b_{m}.$ $(ab)_{m}=a_{m}b_{m}$

The Power of a Product Property has been proven.

A power with an integer exponent $m$ whose base is a ratio of a non-zero numerator $a$ to a non-zero denominator $b$ can be written as the ratio of two powers with bases $a$ and $b$ and the same exponent $m.$

$(ba )_{m}=b_{m}a_{m} $

For the rule to be true for $a=0,$ $m$ must be greater than zero.

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, $(ba )_{m}$ will be rewritten as a product using the definition of a power.

$(ba )_{m}=mtimesba ⋅ba ⋅ba ⋅…⋅ba $

Next, the fractions will be multiplied.
$(ba )_{m}=mtimesb⋅b⋅b⋅…⋅b a⋅a⋅a⋅…⋅amtimes $

Finally, using the definition of a power one more time, the right-hand side of the equation can be written as a ratio of the numerator $a_{m}$ to the denominator $b_{m}.$ $(ba )_{m}=b_{m}a_{m} $

The Power of a Quotient Property has been proven.

If $a$ is a non-zero real number and $n$ is a *positive* integer, then $a$ raised to the power of $-n$ is equal to $1$ over $a$ raised to the power of $n.$

$a_{-n}=a_{n}1 $

It is important to note that the sign of the exponent changes, meaning that the resulting fraction is not simply the reciprocal of the original power.

To prove this identity, the Zero Exponent Property will be considered.

$a_{0}=1,a =0 $

In the above equation, $0$ will be rewritten as $n−n$ and, after some algebraic manipulation and using the Product of Powers Property, the desired result will be obtained. $a_{0}=1$

WriteDiff

Write as a difference

$a_{n−n}=1$

WriteSum

Write as a sum

$a_{n+(-n)}=1$

CommutativePropAdd

Commutative Property of Addition

$a_{-n+n}=1$

SumInExponent

$a_{m+n}=a_{m}⋅a_{n}$

$a_{-n}⋅a_{n}=1$

DivEqn

$LHS/a_{n}=RHS/a_{n}$

$a_{-n}=a_{n}1 ✓$

Any non-zero real number raised to the power of $0$ is equal to $1.$

$a_{0}=1$

To prove this identity, it will be shown that the left-hand side is *always* equal to $1$ for any real number $a.$ To do so, the exponent $0$ will be written as a difference, for instance $b−b.$ Then, the expression will be rearranged using the Quotient of Powers Property.

$a_{0}$

Rewrite $0$ as $b−b$

$a_{b−b}$

DiffInExponent

$a_{m−n}=a_{n}a_{m} $

$a_{b}a_{b} $

QuotOne

$aa =1$

$1✓$