2y^11/3+8y^8/3-10y^5/3=0
To solve the equation 2y^(11/3) + 8y^(8/3) - 10y^(5/3) = 0, you can follow these steps:
Step 1: Factor out the common term.
From the given equation, you can see that each term has a common factor of y^(5/3). Factoring this out, the equation becomes:
y^(5/3)(2y^(6/3) + 8y^(3/3) - 10) = 0
Step 2: Simplify the exponent expressions.
The exponent expressions y^(11/3), y^(8/3), y^(5/3), and y^(6/3) can be simplified. Recall that for any real number a and b, a^(b/c) equals the cth root of a raised to the power of b. Using this property, we can simplify the exponents as follows:
y^(11/3) = (y^(3))^((11/3) / 3) = (y^3)^(11/9)
y^(8/3) = (y^(3))^((8/3) / 3) = (y^3)^(8/9)
y^(5/3) = (y^(3))^((5/3) / 3) = (y^3)^(5/9)
y^(6/3) = (y^(3))^((6/3) / 3) = (y^3)^(2/3)
Step 3: Rewrite the equation.
Using the simplified exponents, we can rewrite the equation as follows:
(y^3)^(11/9) + 8(y^3)^(8/9) - 10(y^3)^(5/9) = 0
Step 4: Let u = y^3.
Substitute u for y^3 in the rewritten equation:
u^(11/9) + 8u^(8/9) - 10u^(5/9) = 0
Step 5: Solve the equation for u.
Now, you have a simplified equation in terms of u. You can solve this equation using algebraic methods or numerical methods such as graphing or approximation techniques. Once you find the values for u, substitute them back into the equation u = y^3 to find the corresponding values for y.
Note: In general, solving equations with fractional exponents may involve advanced techniques, such as factoring, using the quadratic formula, or numerical methods if an analytical solution is not readily available.