Question 8 - of 12 Step 1 of 1
No Time Limit
Solve the following equation by factoring.
\[
5 y^{\frac{14}{5}}-42 y^{\frac{9}{5}}+16 y^{\frac{4}{5}}=0
\]
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2 Points
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Step 1 ：The given equation is a polynomial equation where the variable y is raised to different powers. The powers are in fractional form, but they all have the same denominator of 5. This suggests that we can treat the equation as a quadratic equation in terms of \(y^{\frac{4}{5}}\).

Step 2 ：We can rewrite the equation as: \[5 (y^{\frac{4}{5}})^3 - 42 (y^{\frac{4}{5}})^2 + 16 y^{\frac{4}{5}} = 0\]. This is now in the form of a quadratic equation \(ax^3 - bx^2 + cx = 0\), where \(x = y^{\frac{4}{5}}\).

Step 3 ：We can solve this equation by factoring.

Step 4 ：The solutions for the equation are [0.0, 0.400000000000000, 8.00000000000000, 0.4 - 0.e-20*I, 0.4 - 0.e-20*I, 0.4 + 0.e-20*I, 0.4 + 0.e-20*I, 8.0 - 0.e-19*I, 8.0 - 0.e-21*I, 8.0 + 0.e-21*I, 8.0 + 0.e-19*I]. However, some of these solutions are complex numbers (with an imaginary part), which are not valid solutions for the original equation. We need to filter out the real solutions.

Step 5 ：The real solutions are [0, 0.400000000000000, 8.00000000000000].

Step 6 ：Final Answer: The real solutions to the equation are \(\boxed{0}\), \(\boxed{0.4}\), and \(\boxed{8}\).