To come up with examples of variable expressions using radicals that meet the required conditions, we should consider radicals with odd and even indexes.

Odd index

For odd indexed radicals, we can calculate any radicand, either positive or negative. With this knowledge we can come up with an example of a radical that does not need an absolute value when simplified.

Expression 3 x^{3} = Simplified x

Even index

The only kind of radicals where we have a limitation to what value we can substitute in the radicand is one with an even index. Let's consider a few expressions involving radicals with an even index.

Expression 4 x^{8} x^{2} = = Simplified x^{2} x

In the first equation, we can substitute any value. Values that are positive, negative, or

0

will all work.

$x$	$4 x^{8} = x^{2}$	Simplify
$- 1$	$4 (- 1)^{8} = ? (- 1)^{2}$	$1 = ✓ 1$
$0$	$4 0^{8} = ? 0^{2}$	$0 = ✓ 0$
$1$	$4 1^{8} = ? 1^{2}$	$1 = ✓ 1$

Let's try the same thing with the second equation.

$x$	$x^{2} = x$	Simplify
$- 1$	$(- 1)^{2} = ? - 1$	$1 \neq = - 1$
$0$	$0^{2} = ? 0$	$0 = ✓ 0$
$1$	$1^{2} = ? 1$	$1 = ✓ 1$

As we can see, we cannot substitute a negative value in the second equation. Therefore, we need an absolute value when we simplify it.

x^{2} = ∣ x ∣

Exercise