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1. $(ab)^n = a^nb^n$
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$(2 \cdot 3)^2 = 2^2 \cdot 3^2 = 4 \times 9 = 36$
$(2 \cdot 3)^{-2} = 2^{-2} \cdot 3^{-2} = \dfrac{1}{4} \times \dfrac{1}{9} = \dfrac{1}{36}$
$(4(-3))^{2} = 4^2 \cdot (-3)^2 = 16 \times 9 = 144$
$(xy)^{-4} = x^{-4}y^{-4}$
$(-xy)^3 = (-x)^3(y)^3$
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#[The $n$th power of a product is the product of the $n$th powers.][La $n$ª potencia de un producto es el producto de las $n$ª potencias.]#
#[To see why this rule holds in the case of positive exponents, notice that][Para intender por qué se aplica esta regla en el caso de exponentes positivos, observa que]#
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$(ab)^n$=
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$(ab \cdot ab \cdot ab \ \cdots \ ab )$
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$(a \cdot a\cdot a\ \cdots \ a)(b \cdot b\cdot b\ \cdots \ b)$
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$a^nb^n$
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#[The rule does not apply to sums: ][La regla no se aplica a sumas: ]#
$\qquad (a+b)^n \neq a^n + b^n$
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2. $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
(if $b \neq 0$)
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$\left(\dfrac{3}{2}\right)^4 = \dfrac{3^4}{2^4} = \dfrac{81}{16}$
$\left(\dfrac{x}{y}\right)^{-2} = \dfrac{x^{-2}}{y^{-2}}$
$\left(\dfrac{1}{y}\right)^3 = \dfrac{1^3}{y^3} = \dfrac{1}{y^3}$
$\left(\dfrac{-2}{-3}\right)^2 = \dfrac{(-2)^2}{(-3)^2} = \dfrac{4}{9}$
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#[The $n$th power of a quotient is the quotient of the $n$th powers.][La $n$ª potencia de un cociente es el cociente de las $n$ª potencias.]#
#[To see why this rule holds in the case of positive exponents, notice that][Para intender por qué se aplica esta regla en el caso de exponentes positivos, observa que]#
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$\left(\dfrac{a}{b}\right)^n$=
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$\left(\dfrac{a}{b} \cdot \dfrac{a}{b} \cdot \dfrac{a}{b} \ \cdots \ \dfrac{a}{b} \right)$
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$\dfrac{a \cdot a\cdot a\ \cdots \ a}{b \cdot b\cdot b\ \cdots \ b}$
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$\dfrac{a^n}{b^n}$
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#[The rule does not apply to differences: ][La regla no se aplica a restas: ]#
$\qquad (a-b)^n \neq a^n - b^n$
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