Radicals

A square root is a radical number. 

We use the symbol \sqrt{a} to indicate the ‘square’ root of a.

The cube root of a is written as \sqrt[3]{a} .

a=b^{n},, , , , , b, is , the, n^{th}, root, of, a

  \sqrt[n]{a}=\sqrt[n]{b^{n}}=b

 

If n=2 , , and , , b^{2}=a , the root is a square root.       \sqrt{a}= \sqrt{b^{2}}=b

If n=3, , and , , b^{3}=a , the root is a cube root.         \sqrt[3]{a}=\sqrt[3]{b^{3}}=b

 

           25=5^{2}

 

\sqrt{25}=\sqrt{5^{2}}=5 

 

also\, \, 25=\left ( -5 \right )^{2}

  

 \sqrt{25}=\sqrt{\left ( -5 \right )^{2}}=-5 

\therefore \sqrt{25}\, \, could\, \, be\, \, +5\, \, or\, \, -5

 

 

The Principal n^{th} root is  \sqrt[n]{a} that has the same sign as a.

\sqrt[n]{a}\, \, \, \, \, \, \, \, \, \, \, \, n   is the index of the radical

      a   is called the radicand

 

 \sqrt{4}\neq \pm 2\, \, \, \, \, \, -\sqrt{4}=-2\, \, \, \, and \, \, \, \, \sqrt{4}=2

 

         n^{th}  root of Real Numbers

Real No a                   Positive Integer n               Roots of a

1.\, \, a> 0\, \, or\, \, \left ( + \right )                    n is even                           \sqrt[n]{a},-\sqrt[n]{a}

2.\, \, a> 0\, \, or\, \, a< 0                   n is odd                               \sqrt[n]{a}

3.\, \, a< 0\, \left ( - \right )                         n is even                        no real roots

4., , , a=0                       n is odd or even                   \sqrt[n]{0}=0

Examples of each of the above 4 possible events.

1.\, \, \sqrt[4]{16}=\sqrt[4]{\left ( 2 \right )^{4}}=2,\, \, -\sqrt[4]{16}=-2

2.\, \, \sqrt[3]{8}=2,\, \, \sqrt[3]{-8}=-2

3.\, \, \sqrt{-9}\, \, is\, \, not\, \, a\, \, real\, number

4.\, \, \sqrt[5]{0}=0

 Perfect Squares \rightarrow 1,4,9,16,25,36,49,...

 Perfect Cubes \rightarrow 1,8,27,64,125,. . .

 

 

 

Filed under: Pre-requisites for Pre-Calculus

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