# square root + cube root of a number

• Dec 5th 2010, 10:07 AM
dugongster
square root + cube root of a number
The square root and cube root of a number is equal to 36. What is the number ?

Can this really be solved by logarithms ?

I got far as log ( 36 - y raised to 1/2 ) = log y

I don't know what to do next ?
• Dec 5th 2010, 10:14 AM
e^(i*pi)
$\displaystyle \sqrt{x} + \sqrt[3]{x} = 36 \: \: , \: x \geq 0$

Logarithms seem like more trouble than they're worth.

Let $\displaystyle u = \sqrt[6]{x} \implies x = u^6$ (6 is the LCD of 2 and 3).

Thus we get $\displaystyle f(u) = u^3 + u^2 - 36 = 0$. u^3 comes from the rules of surds, namely: $\displaystyle \sqrt{a} = a^{1/2} = a^{3/6} = \sqrt[6]{a^3} = u^3$

Solve the polynomial for u and then it should be trivial to find the value of x but remember to check for extraneous results and that any solution is in the domain

edit 2: Hint f(3) = 0

If you don't mean adding them then please clarify what the question wants
• Dec 5th 2010, 10:35 AM
dugongster
You're the man !