Solving Exponential Equation @KasyannoEZMath

This tutorial video explains how to solve the given exponential equation.

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[kfjfkeofitorhf]

16^4-256, 4^4=4^4

X=64=3/64=3/4^3=4
X=4
16^4-256×4^4=4^4

[walterwen]

Solving Exponential Equation: 16^cbrt(x) – 255[4^cbrt(x)] = 256
Let y = 4^cbrt(x); y^2 = [4^cbrt(x)]^2 = (4^2)^[cbrt(x)] = 16^cbrt(x)
16^cbrt(x) – 255[4^cbrt(x)] = y^2 – 255y = 256, y^2 – 255y – 256 = 0
(y – 256)(y + 1) = 0; y + 1 = 4^cbrt(x) + 1 > 0
y – 256 = 0; y = 4^cbrt(x) = 256 = 4^4, x^(1/3) = 4; x = 4^3 = 64
Answer check:
16^cbrt(x) – 255[4^cbrt(x)] = 16^4 – 255(4^4) = 4^8 – 255(4^4)
= (4^4)(4^4 – 255) = 256; Confirmed

[gelbkehlchen]

Solution:
16^(³√x)-255*4^(³√x) = 256 ⟹
4^(2*³√x)-255*4^(³√x) = 256 |with u=³√x ⟹
4^(2*u)-255*4^u = 256 |with z=4^u ⟹
z²-255*z = 256 |-256 ⟹
z²-255*z-256 = 0 |p-q-formula ⟹
z1/2 = 255/2±√[(255/2)²+256)] = 255/2±1/2*√[65025+1024] = 255/2±1/2*257 ⟹
z1 = 255/2+1/2*257 = 256 and z2 = 255/2-1/2*257 = -1 ⟹

1.case: 4^u1 = z1 = 256 = 4^4 |because of the same base ⟹ u1 = 4 ⟹
³√x1 = u1 = 4 |()³ ⟹
x1 = 4³ = 64
2.case: 4^u2 = z2 = -1 |that is not defined.
The only solution is x = 64.