Extracting and Inserting Factors from a Root

When dealing with a root that contains a radicand with powers, you can simplify the expression by:

bringing a factor out of the root, as long as its exponent matches or is a multiple of the root's index
moving a factor into the root by multiplying its exponent by the root's index

Let's look at a few practical examples.

Example

In this scenario, we have a square root ($\sqrt{}$) with an index of 2 and a factor $a^2$ inside the radicand.

$$ \sqrt{a^2 b} $$

Since the exponent of $a$ (2) matches the index of the root (2), you can "extract" $a$, reducing its exponent to 1 (i.e., $2/2$). The result is $a \sqrt{b}$.

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

Example

In this example, the root index is 4 (fourth root) and the radicand is $a^8b$.

$$ \sqrt[4]{a^8 b} $$

Because the exponent of $a$ (8) is a multiple of the root's index (4), you can take $a$ out with a reduced exponent: $8/4 = 2$. The result is $a^2 \sqrt[4]{b}$.

$$ \sqrt[4]{a^8 b} = a^2 \sqrt[4]{b} $$

Example

Here, the exponent of $a$ is 9, and the root index is 4.

$$ \sqrt[4]{a^9 b} $$

You can extract $a^8$ (since 8 is a multiple of 4), resulting in $a^2$ outside the root ($8/4 = 2$). What's left under the root is $a b$, so the final result is $a^2 \sqrt[4]{ab}$.

$$ \sqrt[4]{a^9 b} = a^2 \sqrt[4]{ab} $$

Example

In this case, the term $a$ is outside the fourth root ($\sqrt[4]{}$).

$$ a \sqrt[4]{b} $$

To move $a$ inside the root, raise $a$ to the power of the root's index, which is $4$, resulting in $a^4$ inside the root.

$$ a \sqrt[4]{b} = \sqrt[4]{a^4 b} $$

Example

In this more general example, $a^p$ is outside the root.

$$ a^p \cdot \sqrt[4]{b} $$

To bring it into the fourth root, raise $a^p$ to the power of 4, resulting in $a^{4p}$ inside the root. The final result is $\sqrt[4]{a^{4p} b}$.

$$ a^p \cdot \sqrt[4]{b} = \sqrt[4]{a^{4p} b} $$

These techniques are particularly useful for simplifying algebraic expressions with roots and powers, making calculations more efficient and manageable.