x ∈ C, x ≠ 0
y times the cube root of x2y
The domain is all real numbers except when the denominator equals zero: x2 - 4 = 0 x2 = 4 x = 2, -2 So the domain is all real numbers except 2 and -2.
(x2 + x2)=
The domain of y = x2 is [0,+infinity]
Suppose x3 = -64 then x3 + 64 = 0So (x + 4)*(x2 - 4x + 16) = 0So x = 4 or x = 2 +/- 2i*sqrt(3) where i is the imaginary square root of -1.-4 * -4 * -4 = -64So cube root of -64 = -4
1
y = (square root 1- x2) + (cube root x2)
y times the cube root of x2y
1
The domain is all real numbers except when the denominator equals zero: x2 - 4 = 0 x2 = 4 x = 2, -2 So the domain is all real numbers except 2 and -2.
(x2 + x2)=
The domain of y = x2 is [0,+infinity]
Suppose x3 = -64 then x3 + 64 = 0So (x + 4)*(x2 - 4x + 16) = 0So x = 4 or x = 2 +/- 2i*sqrt(3) where i is the imaginary square root of -1.-4 * -4 * -4 = -64So cube root of -64 = -4
square root of (x2 + 1) = no simplification (square root of x2) + 1 = x + 1
It can be determined by solving the equation: x2-x3 = 1 x2(1-x) = 1 x = - 1/(1-x)1/2 (It can be shown by sketch that the root is negative) By iteration, the root is -0.755 (3 s.f.)
Using a radical (square root) bar. I can't get one on the screen, but I'm sure you know what they look like. Example: fractional exponents can be rewritten in radical form: x2/3 means the cube root of (x2) ... write a radical with an index number 3 to show cube root and the quantity x2 is inside the radical. Any fractional exponent can be done the same way. The denominator of the fractional exponent becomes the index of the radical, but the numerator stays as a whole number exponent in the radical.
Personally I prefer to convert roots to fractional powers for this kind of problem. cube root of x squared is x2/3, and square root of x is x1/2. Adding the exponents, you get x2/3 times x1/2 = x7/6, that is, the sixth root of x to the seventh power - where it doesn't matter whether you take the sixth root first, or raise to the sevents power first. Alternatively, you can convert all roots to sixth roots, and multiply - and of course, get the same result.