Math Exponents and roots Partially taking the root

# Partially taking the root

With the help of root rules, it is possible to partially take roots. The root rule is applied backwards:

$\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}$
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### Remember

When partially taking the root, the root is divided into takeable and non-takeable roots. Then the root is taken from the takeable while the non-takeable roots remain.
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### Method

1. Decomposition into prime numbers
2. Summarize into factors with even exponents
3. "Divide" the root (apply the root law backwards)
4. Partially take roots

### Example

$\sqrt{44}$

1. #### Decomposition into prime numbers

Divide the number into a product of primes.
$=\sqrt{2\cdot2\cdot11}$
2. #### Summarize into factors with even exponents

If possible, summarize in factors that have an even exponent.
$=\sqrt{2^2\cdot11}$
3. #### "Divide" the root

Divide the root by applying the multiplication law backwards. The result is a takeable and a non-takeable root.
$=\sqrt{2^2}\cdot\sqrt{11}$
4. #### Partially take roots

Take roots (from the takeable ones) by shortening the exponent and the root.
$=\sqrt[\color{red}{2}]{\color{blue}{2}^{\color{red}{2}}}\cdot\sqrt{11}=\color{blue}{2}\cdot\sqrt{11}$

$\sqrt{44}=2\sqrt{11}$