Answer
The solutions are $x = -2\sqrt {2}, 2\sqrt {2}, -2i\sqrt {2}, \text{ and } 2i\sqrt {2}$.
Work Step by Step
We see that $x^4 - 64 = 0$ is the difference of two squares, so let us factor the equation first to simplify it a little according to the formula $a^2 - b^2 = (a + b)(a - b)$:
$(x^2 - 8)(x^2 + 8) = 0$
We can now use the Zero-Product Property by equating each factor to zero, then solving each equation.
First factor:
$x^2 - 8 = 0$
$x^2 = 8$
We can solve for $x$ by taking the square root of $8$; however, let us simplify that by expressing $8$ as the product of a square and another number:
$x =\pm \sqrt {4(2)}$
Now, we can take the square root of $4$ and remove it from under the radical sign:
$x = \pm 2\sqrt {2}$
Second factor:
$x^2 + 8 = 0$
$x^2 = -8$
We can solve for $x$ by taking the square root of $-8$; however, let us simplify that by expressing $-8$ as the product of a perfect square and another number:
$x =\pm \sqrt {(-4)(2)}$
Now, we can take the square root of $-4$, which is 2i, and remove it from under the radical sign:
$x = \pm 2i\sqrt {2}$
The solutions are $x = \pm 2\sqrt {2}, \pm 2i\sqrt {2}$.
