To find the value of
mn, we start with the general term in the expansion of
(mx+)n, given by the binomial theorem:
Tk+1=()(mx)n−k()kWe need to find the 4 th term in this expansion, so let
k=3. Substituting
k=3 into the general term formula, we get:
T4=()(mx)n−3()3Simplify the expression:
T4=()mn−3xn−3⋅Combine the exponents of
x :
T4=()mn−3xn−3−3=()mn−3xn−6Given that the 4 th term is
, this implies:
()mn−3xn−6=For the term to be a constant (i.e., not dependent on
x ), the exponent of
x must be zero:
n−6=0⇒n=6Substitute
n=6 into the equation for the 4th term:
T4=()m6−3x0= This simplifies to:
()m3=Calculate
() :
()==20Substitute
() into the equation:
20m3=Solve for
m3 :
m3=×==Taking the cube root of both sides, we find:
m=Finally, calculate
mn :
mn=()×6=3