How do you simplify $(7^2/3)^(5/2)$ ?

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Answer 1 · 2018-04-15T12:28:07

$( 7 / sqrt (3) )^ 5$

First, let's set out the exponent rules that are relevant:

(1) $(a ^ b)^c = a ^(bc)$

(2) $(e * f ) ^g = e^g * f^g$

(3) $(1 / h) ^ j = h ^-j$

Now we first use rule (2):

$( 7 ^2 / 3) ^( 5/ 2) = (7 ^2 ) ^ ( 5 / 2 ) * (1 / 3) ^( 5 / 2 )$

We can now use rule (3):

$(7 ^2 ) ^ ( 5 / 2 ) * (1 / 3) ^( 5 / 2 ) = (7 ^2 ) ^ ( 5 / 2 ) * 3 ^( - 5 / 2 )$

We use rule (1) to multiply out the left hand term:

$(7 ^2 ) ^ ( 5 / 2 ) * 3 ^( - 5 / 2 ) = 7 ^( 2 * 5 / 2 ) * 3 ^( - 5 / 2 )$

$= 7 ^ ( 5 ) * 3 ^ (- 5 / 2)$

To simplify further we could again use rule (1) and rule (3):

$7 ^ ( 5 ) * 3 ^ (- 5 / 2) = ( 7 / 3 ^ (1 / 2) )^ 5$

As $x^(1/2) -= sqrt (x)$

$( 7 / 3 ^ (1 / 2) )^ 5 =( 7 / sqrt (3) )^ 5$

How do you simplify (7^2/3)^(5/2)(7^2/3)^(5/2)?