Multiplication involving surds, also known as radicals, follows specific rules to simplify and perform the operation. Surds are expressions that include a radical symbol (√) and a number or variable under the radical. Here's how multiplication of surds works:
\(\sqrt{a}+\sqrt{b}\) ≠ \(\sqrt{ab}\)
\(\sqrt{2}+\sqrt{3}\) ≠ \(\sqrt{5}\)
\(\sqrt{a}.\sqrt{b}\)= \(\sqrt{ab}\)
\(\sqrt{2}.\sqrt{3}\)= \(\sqrt{6}\)
Surds of the same order can be multiplied
\(\sqrt[n]{x}.\sqrt[n]{y}=x^{\frac{1}{n}}.y^{\frac{1}{n}}=(xy)^{\frac{1}{n}}=\sqrt[n]{xy}\)
\(\sqrt[n]{x}.\sqrt[m]{y}=x^{\frac{1}{n}}.y^{\frac{1}{m}}=x^{\frac{m}{nm}}.y^{\frac{n}{nm}}\)
\(=(x^{m})^{\frac{1}{nm}}.(y^{n})^{\frac{1}{nm}}\)=\((x^{m}.y^{n})^{\frac{1}{nm}}\)
=\(\sqrt[mn]{x^{m}.y^{n}}\)
Example:
\(\sqrt[3]{2} \times \sqrt{3}=2^{\frac{1}{3}}.3^{\frac{1}{2}}\)
=\(2^{\frac{1 \times 2}{3 \times 2}}.3^{\frac{1 \times 3}{2 \times 3}}\) [ LCM of 2 &3 is 6]
=\(2^{\frac{2}{6}}.3^{\frac{3}{6}}\)
=\(\sqrt[6]{2^{2}}.\sqrt[6]{3^{3}}\)
=\(\sqrt[6]{4 \times 27}\)
=\(\sqrt[6]{108}\)