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

1 Answer
Apr 15, 2018

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

Explanation:

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 #