Definition of roots and identities.
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| Powers |
Identities for calculating with powers. |
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Roots are special powers: \(\quad
\sqrt[p]{a} = a^{\frac{1}{p}}\).
Note: \(\sqrt{a}\) is only defined
for \(a\geq 0\) and is also greater
than or equal to zero itself.
From the rules for powers we obtain:
\(\displaystyle{\sqrt[q]{\sqrt[p]{a}} =
\sqrt[pq]{a}}\)
\(\displaystyle{\sqrt[p]{ab} =
\sqrt[p]{a} \sqrt[p]{b}}\)
\(\displaystyle{\frac{\sqrt[p]{a}}{\sqrt[p]{b}} =
\sqrt[p]{\frac{a}{b}}}\)
We simply write \(\sqrt{a}\) for the
square root \(\sqrt[2]{a}\). Powers and
roots combine well as the latter is just a special case of the
former:
\(\qquad a^{\frac{p}{q}} = \left(
a^{\frac{1}{q}} \right)^p = \left( \sqrt[q]{a} \right)^p\)
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