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.
    $=\sqrt[\color{red}{2}]{\color{blue}{2}^{\color{red}{2}}}\cdot\sqrt{11}=\color{blue}{2}\cdot\sqrt{11}$

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