Answer
$(a)$ A real number $b$ multiplied by itself $n$ times is equal to $a$.
$(b)$ It is true.
$(c)$ If $n$ is even we have $2$ $n$th root.
If $n$ is odd we have $1$ $n$th root.
$(d)$ $\sqrt[4]{-2}$ is not a real number
$\sqrt[3]{-2}$ is a real number
Work Step by Step
$(a)$
$\sqrt[n]a=b$
The equation above means that a number $b$ multiplied by itself $n$ times is equal to $a$.
$(b)$
$\sqrt{a^2}=|a|$
The equation above is true. Let's see a few example:
$\sqrt{2^2}=|2|$
$\sqrt{4}=2$
$2=2$
$\sqrt{(-2)^2}=|2|$
$\sqrt{4}=2$
$2=2$
$(c)$
If the $n$ is an even number, then it can have $2$ $n$th root. Either with a positive sign or negative sign. It is due to the reason, that a positive $a$ number to the even power gives us the same number as $-a$ to the same even power.
Also shown in $(b)$.
On the other hand, if $n$ is an odd number, then we have only one $n$th root.
$(d)$
$\sqrt[4]{-2}$ is not a real number. We cannot take an even root of a negative number. There is no way to get a negative number by multiplying a negative number by itself even times.
There is no such a real number which to the power even value gives a negative number.
In case of odd root we have real number solution. $\sqrt[3]{-2}$. If we multiply a negative number by itself odd times we will get a negative number.
