Answer
$a)$ $x$-intercepts: $\pm4;$ $y$-intercept: None
$b)$ $x$-intercept: $4;$ $y$-intercept: $8$
Work Step by Step
$a)$ $y=\sqrt{x^{2}-16}$
To find the $x$-intercept, set $y$ equal to $0$ and solve for $x$:
$y=\sqrt{x^{2}-16}$
$0=\sqrt{x^{2}-16}$
$(\sqrt{x^{2}-16})^{2}=0^{2}$
$x^{2}-16=0$
$x^{2}=16$
$x=\pm\sqrt{16}$
$x=\pm4$
To find the $y$-intercepts, set $x$ equal to $0$ and solve for $x$:
$y=\sqrt{x^{2}-16}$
$y=\sqrt{0^{2}-16}$
$y=\pm\sqrt{-16}$
Since solving for $y$ yields a complex solution, this equation does not have $y$-intercept
$b)$ $y=\sqrt{64-x^{3}}$
To find the $x$-intercept, set $y$ equal to $0$ and solve for $x$:
$y=\sqrt{64-x^{3}}$
$0=\sqrt{64-x^{3}}$
$(\sqrt{64-x^{3}})^{2}=0^{2}$
$64-x^{3}=0$
$x^{3}=64$
$x=\sqrt[3]{64}$
$x=4$
To find the $y$-intercept, set $x$ equal to $0$ and solve for $y$:
$y=\sqrt{64-x^{3}}$
$y=\sqrt{64-(0)^{3}}$
$y=\sqrt{64}$
$y=8$