Math Olympiad Problem: Solve `k^{k^3}=36` Step-by-Step

 Math Olympiad Problem `k^{k^3}=36` . Find the Value of `k`

Solution

`k^{k^3}=36`

We can rewrite `36` as

`36 = 6^2`

So the equation becomes

`k^{k^3}=6^2`

Take natural logarithm on both sides

`\ln(k^{k^3})=\ln(36)`

Using the logarithmic rule `\ln(a^b)=b\ln a`

`k^3 \ln k = \ln 36`

Let

`\ln k = x`

Then

`k=e^x`

and

`k^3=e^{3x}`

Substitute into the equation

`e^{3x}\cdot x=\ln 36`

`x e^{3x}=\ln 36`

Let

`y=3x`

Then

`x=\frac{y}{3}`

Substitute

`\frac{y}{3}e^{y}=\ln 36`

Multiply both sides by `3`

`y e^{y}=3\ln 36`

Using Lambert W function

`y=W(3\ln 36)`

Since `y=3x`

`x=\frac{W(3\ln 36)}{3}`

But `x=\ln k`

`\ln k=\frac{W(3\ln 36)}{3}`

Exponentiate both sides

`k=e^{\frac{W(3\ln 36)}{3}}`

Numerically

`k \approx 1.85`

Answer

`k \approx 1.85`