How can I simplify #((f^-16)/(256g^4h^-4))^(-1/4)#?

2 Answers
Dec 14, 2015

#(4gf^2)/h#

Very slightly different approach.

Explanation:

Given: #((f^(-16))/(256g^4h^(-4)))^(-1/4)#

The trick is to remember that anything raised to a negative power is inverted

Rewrite as:

#((h^4)/(256g^4f^16))^(-1/4)#

Rewrite as:

#((256g^4f^16)/h^4)^(1/4)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For method think of: #sqrt(2^2xx2^2) = 4 = sqrt(2^2) xx sqrt(2^2)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Giving: #(root4(256)xxroot4(g^4)xxroot4(f^16))/(root4(h^4))#

#(4gf^2)/h#

Dec 14, 2015

#((f^(-16))/(256g^4h^(-4)))^(-1/4) = abs((4f^4g)/h)#

Explanation:

If #a, b, c > 0# then #(a^b)^c = a^(bc)#

Also #a^(-b) = 1/(a^b)#

Hence we find:

#((f^(-16))/(256g^4h^(-4)))^(-1/4)=((256g^4h^(-4))/(f^-16))^(1/4)=((4^4f^16g^4)/(h^4))^(1/4)#

#=((4^4(f^4)^4g^4)/(h^4))^(1/4)=(((4f^4g)/h)^4)^(1/4)=abs((4f^4g)/h)#

If #a > 0# then #a^(1/4)# denotes the Real positive fourth root.

Hence for any #a in RR# we have #root(4)(a^4) = abs(a)#