sixthroot x^5 = 9 * twelfthroot x^7 - 14*thirdroot x
x^5/6 = 9x^7/12 - 14x^1/3
x^10/12 - 9x^7/12 + 14x^4/12 = 0
x^4/12(x^6/12 - 9x^3/12 + 14) = 0
x^1/3 (x^1/4 -7)(x^1/4 - 2) = 0
x = 0,7^4,2^4
but if you plug in all 3 answers back to check for extraneous solutions, are any extraneous?
To solve the equation sixthroot(x^5) = 9 * twelfthroot(x^7) - 14 * thirdroot(x), we can use the rules of exponents and the properties of roots.
First, let's simplify each side of the equation separately.
1. Simplifying the left side:
The sixth root of x^5 is the same as raising x^5 to the power of 1/6. So, the left side of the equation becomes x^(5/6).
2. Simplifying the right side:
The twelfth root of x^7 is the same as raising x^7 to the power of 1/12. So, the first term on the right side becomes x^(7/12).
The third root of x is the same as raising x to the power of 1/3. So, the second term on the right side becomes x^(1/3).
Therefore, the right side of the equation becomes 9 * x^(7/12) - 14 * x^(1/3).
Now, our equation is x^(5/6) = 9 * x^(7/12) - 14 * x^(1/3).
To proceed further, we need to equate the exponents.
5/6 = 7/12 - 1/3
To simplify this equation, let's find a common denominator for the fractions on the right side.
5/6 = 7/(2 * 6) - 2/(2 * 3)
Simplifying further:
5/6 = 7/12 - 4/6
Now, we can combine the fractions:
5/6 = (7 - 4 * 2) / 12
5/6 = (7 - 8) / 12
5/6 = -1/12
The equation is not valid since 5/6 is not equal to -1/12.
Therefore, there is no solution to the given equation sixthroot(x^5) = 9 * twelfthroot(x^7) - 14 * thirdroot(x).