Fundamentals Of Surds
Laws of radicals :
1. If \(
\sqrt[n]{a}
\) and \(
\sqrt[n]{b}
\) are two radicals of same order, then \(
\sqrt[n]{a}
\) x\(
\sqrt[n]{b}
\) = \(
\sqrt[n]{{ab}}
\)
Ex : \(
\sqrt[4]{3} \times \sqrt[4]{2} = \sqrt[4]{{3 \times 2}} = \sqrt[4]{6}
\)
2. If \(
\sqrt[n]{a}
\) and \(
\sqrt[n]{b}
\) are two radicals of same order. 'n' then \(
\frac{{\sqrt[n]{a}}}
{{\sqrt[n]{b}}} = \sqrt[n]{{\frac{a}
{b}}}
\)
Ex: \(
\frac{{\sqrt[3]{{10}}}}
{{\sqrt[3]{3}}} = \sqrt[3]{{\frac{{10}}
{3}}}
\)
3. If m and n are two natural numbers, then for any positive rational numbers a we have \(
\sqrt[n]{{\sqrt[m]{a}}} = \sqrt[{mn}]{a}
\)
Ex : \(
\sqrt[4]{{\sqrt[3]{6}}} = \sqrt[{12}]{6}
\)
4. If m,n are two natural numbers and 'a' is any positive rational number then \(
\sqrt[n]{{\sqrt[m]{{(a^P )^m }}}} = \sqrt[n]{{\sqrt[m]{{a^{pm} }}}} = \sqrt[n]{{a^p }}
\)
Ex :\(
\sqrt[6]{{729}} = \sqrt[{2 \times 3}]{{729}} = \sqrt[2]{{\sqrt[3]{{9^3 }}}} = \sqrt[2]{9} = 3
\)
Example 1: Convert \(
\frac{2}
{3}\sqrt 5
\) into pure surd
Sol : \(
\frac{2}
{3}\sqrt 5 = \sqrt[{}]{{\frac{4}
{9}.5}} = \sqrt {\frac{{20}}
{9}}
\)
Example 2: Convert \(
\sqrt {\frac{{80}}
{{11}}}
\) into mixed surd
Sol : \(
\sqrt {\frac{{80}}
{{11}}} = \sqrt {\frac{{16 \times 5}}
{{11}}} = 4.\sqrt {\frac{5}
{{11}}}
\)
Example 3 : Convert \(
\sqrt {\frac{{50}}
{4}}
\) into its simplest form
Sol : \(
\sqrt {\frac{{50}}
{4}} = \sqrt {\frac{{25}}
{2}} = \frac{5}
{{\sqrt 2 }} = \frac{5}
{2}\sqrt 2
\)
Note: The product of two similar quadratic surds is a rational number
1) \(
\left( {3\sqrt 3 } \right)\left( {5\sqrt 3 } \right) = 15 \times 3 = 45 \in R
\)
2) The quotient of two similar surds is a rational number
Ex:\(
7 \times \sqrt[3]{4} \div 2 \times \sqrt[3]{4} = \frac{7}
{2} \in R
\)
Uses of laws of radicals : By using the laws of radicals, we can
i) convert a pure surd into a mixed surd
ii) convert a mixed surd into pure surd
iii) simplify the given surds
iv) reduce two given surds to the same order
v) compare the given two surds.
Ex:i)\(
2\sqrt 5 ,3\sqrt 5 ,5\sqrt 5
\) are similar surds
ii)\(
2\sqrt 3 ,2\sqrt 5 ,7\sqrt 2
\) are dissimilar surds.