Radical Numbers – Part II

Properties of Radicals

1. \, \, \sqrt[n]{a^{m}}=\left ( \sqrt[n]{a} \right )^{m}

2.\, \, \sqrt[n]{a}*\sqrt[n]{b}=\sqrt[n]{ab}

3.\, \, \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}},\, b\neq 0

4.\, \, \sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}

5.\, \, \left ( \sqrt[n]{a} \right )^{n}=a

6.\, \, For\, \, n\, \, even,\, \, \sqrt[n]{a^{n}}=\left | a \right |\, \, and\, \, for \, \, n\, \, odd,\, \, \sqrt[n]{a^{n}}=a

Simplifying Radicals

Definition of a radical in Simplest Form:

  • all possible factors have been removed from the radical

  • all rationals [fractions] have radical-free demonimators

  • the index of the radical is reduced

Examples of Simplifying with Even Roots:

a)\, \, \sqrt[4]{48}=\sqrt[4]{16*3}=\sqrt[4]{2^{4}*3}=2\sqrt[4]{3}

b)\, \, \sqrt{75x^{3}}=\sqrt{3*25*x^{2}*x}=5x\sqrt{3x}

c)\, \, \sqrt[4]{\left ( 5x \right )^{4}}=\left | 5x \right |=5\left | x \right |

Simplifying with Odd Roots:

a)\, \, \sqrt[3]{24}=\sqrt[3]{8*3}=2\sqrt[3]{3}

b)\, \, \sqrt[3]{24a^{4}}=\sqrt[3]{8*3*a^{3}*a}=2a\sqrt[3]{3a}

c)\, \, \sqrt[3]{-40x^{6}}=\sqrt[3]{-8x^{6}*5}=-2x^{2}\sqrt[3]{5}

Like Radicals have the same index and the same radicand.

a)\, \, 2\sqrt{48}-3\sqrt{27}                                  b) \, \, \sqrt[3]{16x}-\sqrt[3]{54x^{4}}

\, \, 2\sqrt{16*3}-3\sqrt{9*3}                                  \sqrt[3]{8*2x}-\sqrt[3]{27*2x^{3}*x}

4*2\sqrt{3}-3*3\sqrt{3}                                   2\sqrt[3]{2x}-3x\sqrt[3]{2x}

8\sqrt{3}-9\sqrt{3}                                            \left ( 2-3x \right )\sqrt[3]{2x}

-\sqrt{3}                                                    do not need \left | x \right | because \sqrt[3]{\, \, \, } [odd index]

Rationalizing Denominators and Numerators

To rationalize a radical, you multipy by the conjugate.  a-b\sqrt{m}\, \, \, and \, \, \, a+b\sqrt{m}  are conjugates.  If you multiply a number by its conjugate the result does not have a radical.

\left ( a-b\sqrt{m} \right )\left ( a+b\sqrt{m} \right )=a^{2}-b^{2}\sqrt{m^{2}}=a^{2}-b^{2}m

Examples:

‘D’       \frac{5}{2\sqrt{3}}=\frac{5}{2\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}}=\frac{5\sqrt{3}}{2*3}=\frac{5\sqrt{3}}{6}

‘D’      \frac{2}{\sqrt[3]{5}}=\frac{2}{\sqrt[3]{5}}*\frac{\sqrt[3]{5^{2}}}{\sqrt[3]{5^{2}}}=\frac{2\sqrt[3]{25}}{\sqrt[3]{5^{3}}}=\frac{2\sqrt[3]{25}}{5}

‘D’      \frac{2}{3+\sqrt{7}}=\frac{2}{3+\sqrt{7}}*\frac{3-\sqrt{7}}{3-\sqrt{7}}=\frac{6-2\sqrt{7}}{9-7}=\frac{2\left ( 3-\sqrt{7} \right )}{2}=3-\sqrt{7}

 

‘N’        \frac{\sqrt{5}-\sqrt{7}}{2}=\frac{\sqrt{5}-\sqrt{7}}{2}*\frac{\sqrt{5}+\sqrt{7}}{\sqrt{5}+\sqrt{7}}=\frac{5-7}{2\left ( \sqrt{5}\right )+\sqrt{7}}=\frac{-2}{2\left ( \sqrt{5} -\sqrt{7}\right )}=\frac{-1}{\sqrt{5}+\sqrt{7}}

Filed under: Pre-requisites for Pre-Calculus

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