Calculating with square roots

Algebra: Calculating with exponents and roots

Calculating with square roots

The following rules are very convenient when simplifying expressions with square roots.

$\sqrt{\blue a \cdot \green b}=\sqrt{\blue a} \cdot \sqrt{\green b}$

Example

$\begin{array}{rcl}\sqrt{\blue 4 \cdot \green x}&=&\sqrt{\blue 4} \cdot \sqrt{\green x}\\ &=& 2 \cdot \sqrt{\green x} \end{array}$

Not for adding

There is no similar rule for adding square roots. Hence:

$\sqrt{a+b} \; \red{\ne} \; \sqrt{a} + \sqrt{b}$

Example

$\sqrt{9+16} \;\red{\not=}\; \sqrt{9} + \sqrt{16}$

$\sqrt{\frac{\blue a}{\green b}}=\frac{\sqrt{\blue a}} {\sqrt{\green b}}$

Example

$\begin{array}{rcl}\sqrt{\dfrac{\blue 4}{\green x}}&=&\dfrac{\sqrt{\blue 4}}{\sqrt{\green x}}\\ &=& \dfrac{2}{\sqrt{\green x}} \end{array}$

For $\blue a \geq 0$ we have:

$\sqrt{\blue a^2}=\blue a$

and

$\left(\sqrt{\blue a}\right)^2=\blue a$

Examples

$\begin{array}{rcl}\sqrt{\blue x^4}&=&\sqrt{\blue x^2 \cdot \blue x^2}\\ &=& \sqrt{\blue x^2} \cdot \sqrt{\blue x^2} \\ &=& \blue x \cdot \blue x \\&=&\blue x^2\\ \\ \left(\sqrt{\blue x}\right)^4 &=& \left(\left(\sqrt{\blue x}\right)^2\right)^2 \\ &=& \blue x^2\end{array}$

In the first rule, when $\blue a$ is negative, we will also have a positive number under the root sign. Since a square root is always positive, $\blue a$ cannot be negative.

Because $\sqrt{(-5)^2} =5 \ne -5$ .

We can formulate this rule for negative $\blue a$ , we then find:

$\sqrt{\blue a^2}=\abs{\blue a}$

Here, $\abs{\blue a}$ is the absolute value of $\blue a$ , which means the value is always positive. Hence, $\abs{-5}=5$ .

Assume that $b \ge 0$ and $a \ge 0$ . Write without a square root: $\sqrt{36 \cdot b^{4} \cdot a^{6}}$

$6 \cdot b^2 \cdot a^3$

$\begin{array}{rcl} \sqrt{36 \cdot b^{4} \cdot a^{6}}&=& \sqrt{36} \cdot \sqrt{b^{4}} \cdot \sqrt{a^{6}} \\ && \phantom{xxx}\blue{\text{rule }\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}\\ &=& 6 \cdot b^2 \cdot a^3\\ && \phantom{xxx}\blue{\text{rule }\sqrt{a^2}=a \text{ and square root calculated }} \end{array}$

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