check for extraneous solutions. 0, 7^4 2^4, inside the equation x^5/6 = 9x^7/12 - 14x^1/3
the solutions you plug in are 0, 7^4, and 2^4
well, 0 obviously works
(2^4)^5/6 = 2^10/3 = 2^3*2^1/3 = 8*2^1/3
9(2^4)^7/12 = 9*2^7/3 = 36*2^1/3
14(2^4)^1/3 = 14*2^4/3 = 28*2^1/3
8=36-28
looks good to me
You can check 7^4 in a similar manner.
To check for extraneous solutions, we need to substitute the given values into the equation and see if they satisfy the equation. Here's how you can do it step by step:
1. Start with the equation: x^(5/6) = 9x^(7/12) - 14x^(1/3).
2. Substitute x = 0 into the equation:
0^(5/6) = 9*0^(7/12) - 14*0^(1/3).
This simplifies to 0 = 0 - 0.
Since 0 = 0 is true, x = 0 is a valid solution and not an extraneous solution.
3. Substitute x = 7^4 into the equation:
(7^4)^(5/6) = 9*(7^4)^(7/12) - 14*(7^4)^(1/3).
Simplify each term:
7^(4*(5/6)) = 9*7^(4*(7/12)) - 14*7^(4*(1/3)).
7^(20/6) = 9*7^(14/6) - 14*7^(4/3).
Now, we can simplify 7^(20/6), 7^(14/6), and 7^(4/3) as follows:
7^(10/3) = 9*7^(7/3) - 14*7^(4/3).
At this point, we can see that this is not a valid equation since different powers of 7 on both sides cannot be equal.
Thus, x = 7^4 is an extraneous solution.
4. Substitute x = 2^4 into the equation:
(2^4)^(5/6) = 9*(2^4)^(7/12) - 14*(2^4)^(1/3).
Simplify each term:
2^(4*(5/6)) = 9*2^(4*(7/12)) - 14*2^(4*(1/3)).
2^(20/6) = 9*2^(14/6) - 14*2^(4/3).
Similarly, simplify the powers of 2:
2^(10/3) = 9*2^(7/3) - 14*2^(4/3).
Now, this equation is valid and true. Therefore, x = 2^4 is a valid solution and not an extraneous solution.
In summary, the extraneous solution is x = 7^4, while the valid solution is x = 2^4.