Math Exponents and roots Root rules

Root rules

Root rules regulate how roots behave if you use multiplication, division, exponentiation and root extraction.



The root rules do not apply to adding and subtracting.
  1. The multiplication of roots

    $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{a\cdot b}$
  2. The division of roots

  3. The exponentiation of roots

  4. Root extraction

    $\sqrt[m]{\sqrt[n]{a}}=\sqrt[m \cdot n]{a}$


  • $\sqrt[3]{8}\cdot\sqrt[3]{27}=\sqrt[3]{8\cdot 27}$ $=\sqrt[3]{216}=6$

  • $\frac{\sqrt{8}}{\sqrt{32}}=\sqrt{\frac{8}{32}}$ $=\sqrt{\frac{1}{4}}=\frac{1}{2}$

  • $(\sqrt{2})^4=\sqrt{2^4}$ $=\sqrt{16}=4$

  • $\sqrt{\sqrt{16}} = \sqrt[2 \cdot 2]{16}$ $=\sqrt[4]{16}=2$