Partial rooting

Partial root extraction (coll .: partial root extraction or partial root extraction ) describes a way of reshaping roots (radix). The expression under the root is broken down into its factors so that the factors can be considered individually by applying the laws of the roots . The root is then taken from one or more factors.

Example: ${\displaystyle {\sqrt {8}}={\sqrt {2^{2}\cdot 2}}=2{\sqrt {2}}}$

General consideration

Roots with the same root exponent ( ) are multiplied by rooting the product of the radicands ( ) with this exponent: ${\displaystyle n}$ ${\displaystyle p\cdot q}$

${\displaystyle {\sqrt[{n}]{p}}\cdot {\sqrt[{n}]{q}}={\sqrt[{n}]{p\cdot q}}}$

Conversely, the root can then be extracted from a product factor-wise, one then speaks of partial or partial root extraction :

${\displaystyle {\sqrt[{n}]{p^{n+m}}}={\sqrt[{n}]{p^{n}}}\cdot {\sqrt[{n}]{p^{m}}}=p\cdot {\sqrt[{n}]{p^{m}}}}$

Further examples

${\displaystyle {\sqrt[{3}]{16a^{3}b^{4}}}={\sqrt[{3}]{8}}\cdot {\sqrt[{3}]{2}}\cdot {\sqrt[{3}]{a^{3}}}\cdot {\sqrt[{3}]{b^{3}}}\cdot {\sqrt[{3}]{b}}=2ab\cdot {\sqrt[{3}]{2b}}}$

${\displaystyle {\sqrt {8a^{2}+16a+8}}={\sqrt {8(a^{2}+2a+1)}}={\sqrt {8(a+1)^{2}}}={\sqrt {4}}\cdot {\sqrt {2}}\cdot {\sqrt {(a+1)^{2}}}=2{\sqrt {2}}\cdot |a+1|}$

literature

• Walter Gellert, Herbert Kästner , Siegfried Neuber (Hrsg.): Fachlexikon ABC Mathematik . VEB, Harri Deutsch, Thun and Frankfurt / Main 1978, ISBN 3-87144-336-0 , p. 598 f .