Solve this equation!?

Question

(x^2-x-22)^(4/3)=16

Explain how you got the answer!

Answer 1

get rid of the exponent

x^2 - x - 22 = 8

then simplify

x^2 - x- 30 =0

then factor and solve

Answer 2

Using the calculator,

16 ^ (3/4) = 8

[Press 16 (x^y) key, ( 3/4) = 8]

x^2 -x -22 = 8

x^2 - x -30 = 0

(x + 5)(x - 6) = 0

Answer 3

Raise both sides to the 3/4 power:

x^2-x-22=16^(3/4)

Here, it's good to know that 16=(2^4), so 16^(1/4)=2, so 16^(3/4)=2^3

x^2-x-22=8

x^2-x-30=0

(x-6)(x+5)=0

x=6 or x=-5

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Answer 4

This type of equation involves the the squareroot principle, which is basically the exponentiating of both sides of the equation with the form: u^n = c.  Let u = x^2 - x - 22 && n = 4/3; c = 16.  The exponent n = 4 / 3 is compliant to the same rules of arithmetic, which thus, entails the inverse of numbers.  By taking the reciprocal of the exponent 4 / 3 on both sides of the equation, you have x^2 - x - 22 = (16)^3/4.  The easy way to compute this is by knowing the relationship between numbers such as 16 being a perfect square, etc.  Since 16 is a perfect square; that is, 16 = 4^2 || 4 * 4, any power of 16 can be represented as 4^2*n.  Thus, 16^3/4 = 4^2*3/4 = 2^3 = 8.  Therefore, the equation simplifies to x^2 - x - 22 = 8  The quadratic equation is now generalized to be x^2 - x - 30 = 0.  By factoring, the equation, you have two solutions of x =  6, -5.

J.C