Assuming the problem is:
x = (8^(2/3) * 32^(-5/2))x=(823⋅32−52)
First, use this rule of exponents to rewrite each term on the right side:
x^(color(red)(a) xx color(blue)(b)) = (x^color(red)(a))^color(blue)(b)xa×b=(xa)b
x = (8^(2 xx 1/3) * 32^(-5 xx 1/2))x=(82×13⋅32−5×12)
x = ((8^2)^(1/3) * (32^-5)^(1/2))x=((82)13⋅(32−5)12)
Next, use this rule of exponents to rewrite the 3232 term:
x^color(red)(a) = 1/x^color(red)(-a)xa=1x−a
x = ((8^2)^(1/3) * ((1/32^(- -5))^(1/2))x=((82)13⋅((132−−5)12)
x = ((8^2)^(1/3) * ((1/32^5)^(1/2))x=((82)13⋅((1325)12)
x = ((64)^(1/3) * ((1/33554432)^(1/2))x=((64)13⋅((133554432)12)
We can then use this rule to rewrite the exponents:
x^(1/color(red)(n)) = root(color(red)(n))(x)x1n=n√x
x = root(3)(64) * 1/root(2)(33554432)x=3√64⋅12√33554432
x = root(3)(64) * 1/sqrt(33554432)x=3√64⋅1√33554432
x = root(3)(64) * 1/sqrt(16777216 * 2)x=3√64⋅1√16777216⋅2
Now, use this rule to rewrite the denominator of the fraction:
sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))√a⋅b=√a⋅√b
x = root(3)(64) * 1/(sqrt(16777216 * 2)x=3√64⋅1√16777216⋅2
x = 4 * 1/(sqrt(16777216)sqrt(2))x=4⋅1√16777216√2
x = 4 * 1/(4096sqrt(2))x=4⋅14096√2
x = 4/(4096sqrt(2))x=44096√2
x = 1/(1024sqrt(2))x=11024√2
Next, we can rationalize the fraction:
x = sqrt(2)/sqrt(2) * 1/(1024sqrt(2))x=√2√2⋅11024√2
x = (sqrt(2) * 1)/(sqrt(2) * 1024sqrt(2))x=√2⋅1√2⋅1024√2
x = sqrt(2)/(1024(sqrt(2))^2)x=√21024(√2)2
x = sqrt(2)/(1024 * 2)x=√21024⋅2
x = sqrt(2)/(2048)x=√22048
Or, approximately:
x = 0.00069x=0.00069