Exercises: Terms with Square Roots

Determine the domain of definition

First determine the domain of definition.

$5\cdot \sqrt{2x}\cdot \sqrt{18x}$

Only positive expressions are allowed under the root, therefore only positive values or 0 may be used for $x$.

$D={ℝ}_0^{+}$

Write the term without square roots

Conclude the roots with the calculation rules for square roots.

Determine the domain of definition

First determine the domain of definition.

$${\displaystyle \frac{\sqrt{2a^2}\cdot \sqrt{12a}}{\sqrt{8a}}}$$

Only positive expressions are allowed under the root.

The first root in the numerator is always positive because the variable $a$ is squared.

However, in the second root of the numerator you may only insert positive values or 0 for $a$.

The same applies to the root in the denominator, but here $a$ must be greater than zero.

Therefore only positive values or 0 may be used for $a$.

$D={ℝ}_{}^{+}$, also $a>0$

Write the term without square roots

Conclude the roots with the calculation rules for square roots.

Determine the domain of definition

First determine the domain of definition.

${\left(\sqrt{d-2}\right)}^2$

Only positive expressions are allowed under the root. This gives you the following condition:

$d-2\ge 0$

$d\ge 2$

$D=[2;\infty )$

Write the term without square roots

Take the square root and do not forget the absolute value.

Determine the domain of definition

First determine the domain of definition.

$\sqrt{{(d-2)}^2}$

Only positive expressions are allowed under the root. Since the term under the root is squared, it will always be positive. So there are no definition problems, no matter which values you use for $d$.

$D=ℝ$

Write the term without square roots