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#[Rule][Regla]#
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#[Examples][Ejemplos]#
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#[Comments][Comentarios]#
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1. $a^ma^n = a^{m+n}$
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$2^32^2 = 2^{3+2} = 2^5 = 32$
$2^32^{-2} = 2^{3-2} = 2^1 = 2$
$(-9)^4(-9)^{-2} = (-9)^{4-2} = (-9)^2 = 81$
$x^3x^4 = x^{3+4} = x^7$
$x^{-4}x^3 = x^{-4+3} = x^{-1} = \dfrac{1}{x}$
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#[If the bases in a product match, add the exponents.
If the bases do not match the rule does not apply.][Si las bases en un producto son iguales, suma los exponentes.
Si no son igulaes las bases, la regla no se aplica.]#
#[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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$a^na^m$
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$= (a \cdot a\cdot a\ \cdots \ a)(a \cdot a\cdot a\ \cdots \ a)$
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$= a \cdot a\cdot a\ \cdots \ \cdots \ a$
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#[The rule does not apply to sums: ][La regla no se aplica a sumas: ]#
$\qquad \quad (a+b)^n \neq a^n+b^n$
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2. $\dfrac{a^m}{a^n} = a^{m-n}$
(#[if][si]# $a \neq 0$)
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$\dfrac{2^3}{2^2} = 2^{3-2} = 2^1 = 2$
$\dfrac{x^3}{x^4} = x^{3-4} = x^{-1} = \dfrac{1}{x}$
$\dfrac{2^3}{2^{-2}} = 2^{3-(-2)} = 2^5 = 32$
$\dfrac{x^{-4}}{x^{-3}} = x^{-4-(-3)} = x^{-1} = \dfrac{1}{x}$
$\dfrac{1}{x^{-3}} = \dfrac{x^0}{x^{-3}} = x^{0-(-3)} = x^3$
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#[If the bases in a quotient match, subtract the exponents.
If the bases do not match the rule does not apply.][Si las bases en una cociente son iguales, resta los exponentes.
Si no son igulaes las bases, la regla no se aplica.]#
#[Notice that this rule follows from cancellation in the case of positive exponents.][Observa que esta regla sigue de cancelación en el caso de exponentes positivos.]#
#[The rule does not apply to differences: ][La regla no se aplica a diferencias: ]#
$\qquad a^m-a^n \neq a^{m-n}$
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3. $\dfrac{1}{a^n} = a^{-n}$
(#[if][si]# $a \neq 0$)
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$\dfrac{1}{2^3} = 2^{-3}$
$5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}$
$\dfrac{1}{5^-2} = 5^{-(-2)} = 5^2 = 25$
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#[See "Negative and zero exponents" above.][Ve "Exponentes negativos y cero " arriba.]#
#[Rule 3 is actually a special case of Rule 2:][Regla 3 es en realidad un caso especial de la Regla 2:]#
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$=\dfrac{a^0}{a^n}$ | |
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$=a^{0-n}$ | |
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$=a^{-n}$ | |
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4. $(a^n)^m = a^{nm}\ $
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$(2^3)^2 = 2^{3\times 2} = 2^{6} = 64$
$(x^3)^4 = x^{3 \times 4} = x^{12}$
$(2^{-3})^2 = 2^{(-3)\times2} = 2^{-6} = \dfrac{1}{64}$
$(x^{-3})^{-4} = x^{(-3)\times(-4)} = x^{12}$
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#[Raising a power to a power corresponds to mutliplying the powers.][Elevar una potencia a una potencia corresponde a multiplicar las potencias.]#
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