write the substirution that could be used to make each equation quadratic in u.
For w^1/3- 3w^1/6+8=0,use w=
To make the equation quadratic in u, we need to substitute a new variable for w. Let's use the substitution w = u^2.
To determine the substitution, follow these steps:
Step 1: Let w = u^n, where n is a positive integer. We want to choose a value of n such that when we substitute w = u^n into the equation, it becomes quadratic in u.
Step 2: Substitute w = u^n into the original equation.
Original equation: w^(1/3) - 3w^(1/6) + 8 = 0
Substituting w = u^2:
(u^2)^(1/3) - 3(u^2)^(1/6) + 8 = 0
Simplifying the exponents:
u^(2/3) - 3u^(1/3) + 8 = 0
Now we have a quadratic equation in terms of u, which is what we wanted. So the substitution w = u^2 is the appropriate choice to make the equation quadratic in u.