How do you evaluate #25^ { \frac { 1} { 4} } \cdot 25^ { \frac { - 7} { 4} }#?

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Answer 1 · 2018-02-01T16:11:21

See a solution process below:

First, use this rule of exponents to combine the two terms:

#x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))#

#25^color(red)(1/4) * 25^color(blue)(-7/4) =>#

#25^(color(red)(1/4) + color(blue)(-7/4)) =>#

#25^(color(red)(1/4) - color(blue)(7/4)) =>#

#25^(-6/4) =>#

#25^(-3/2)#

Next, use this rule of exponents to rewrite the expression:

#x^(color(red)(a) xx color(blue)(b)) = (x^color(red)(a))^color(blue)(b)#

#25^(-3/2) =>#

#25^(1/2 xx -3) =>#

#(25^(1/2))^-3 =>#

#(sqrt(25))^-3 =>#

#5^-3#

Now, use this rule of exponents to complete the evaluation:

#x^color(red)(a) = 1/x^color(red)(-a)#

#5^color(red)(-3) =>#

#1/5^color(red)(- -3) =>#

#1/5^color(red)(3) =>#

#1/(5 * 5 * 5) =>#

#1/(25 * 5) =>#

#1/125#