Division of radicals

Example 1: Rationalize the denominator $\dfrac{5x}{\sqrt{7}}$

Solution:

$\dfrac{5x}{\sqrt{7}}$

Multiplying the numerator and denominator by $\sqrt{7}$,

$=\dfrac{5x\cdot \sqrt{7}}{\sqrt{7}\cdot \sqrt{7}}$

$=\dfrac{5x\sqrt{7}}{7}$

Example 2: Rationalize the denominator $\sqrt{\dfrac{3y}{11}}$

Solution:

$\sqrt{\dfrac{3y}{11}}=\dfrac{\sqrt{3y}}{\sqrt{11}}$

Multiplying the numerator and denominator by $\sqrt{11}$,

$=\dfrac{\sqrt{3y}\cdot \sqrt{11}}{\sqrt{11}\cdot \sqrt{11}}$

$=\dfrac{\sqrt{33y}}{11}$

Example 3: Rationalize the denominator $\dfrac{4\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$

Solution:

$\dfrac{4\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$

Multiplying the numerator and denominator by the conjugate of $\sqrt{7}+\sqrt{5}$,

$=\dfrac{(4\sqrt{7}-\sqrt{5})\cdot (\sqrt{7}-\sqrt{5}) }{(\sqrt{7}+\sqrt{5})\cdot (\sqrt{7}-\sqrt{5})}$

$=\dfrac{4\sqrt{7}\cdot \sqrt{7}-4\sqrt{7}\cdot \sqrt{5}-\sqrt{5}\cdot \sqrt{7}+\sqrt{5}\cdot\sqrt{5}}{7-5}$

$=\dfrac{4\cdot 7-4\sqrt{35}-\sqrt{35}+5}{7-5}$

$=\dfrac{33-5\sqrt{35}}{2}$

Example 4: Rationalize the denominator $\dfrac{8\sqrt{6}+4}{\sqrt{6}-2}$

Solution:

$\dfrac{8\sqrt{6}+4}{\sqrt{6}-2}$

Multiplying the numerator and denominator by the conjugate of $\sqrt{6}-2$,

$=\dfrac{(8\sqrt{6}+4)\cdot (\sqrt{6}+2)}{(\sqrt{6}-2)\cdot (\sqrt{6}+2)}$

$=\dfrac{8\sqrt{6}\cdot \sqrt{6}+8\sqrt{6}\cdot 2+4 \cdot \sqrt{6}+ 4 \cdot 2}{6-2^2}$

$=\dfrac{8\cdot 6+16\sqrt{6}+4\sqrt{6}+ 8}{6-4}$

$=\dfrac{48+20\sqrt{6}+ 8}{6-4}$

$=\dfrac{56+20\sqrt{6}}{2}$

$=28+10\sqrt{6}$

Example 5: Rationalize the denominator $\dfrac{17}{2\sqrt{y}-5}$

Solution:

$\dfrac{17}{2\sqrt{y}-5}$

Multiplying the numerator and denominator by the conjugate of $2\sqrt{y}-5$,

$=\dfrac{17\cdot (2\sqrt{y}+5) }{(2\sqrt{y}-5)(2\sqrt{y}+5)}$

$=\dfrac{17\cdot 2\sqrt{y}+17\cdot 5}{(2\sqrt{y})^2-5^2}$

$=\dfrac{34\sqrt{y}+85 }{4y-25}$

Example 6: Rationalize the denominator $2\:\sqrt[3]{\dfrac{13}{7}}$

Solution:

$2\:\sqrt[3]{\dfrac{13}{7}}$

Multiplying the numerator and denominator by two factors of $7$,

$=2\:\sqrt[3]{\dfrac{13\cdot 7 \cdot 7}{7\cdot 7\cdot 7}}$

$=\dfrac{2\:\sqrt[3]{13\cdot 7 \cdot 7}}{\sqrt[3]{7\cdot 7\cdot 7}}$

$=\dfrac{2\:\sqrt[3]{13\cdot 7 \cdot 7}}{7}$

$=\dfrac{2\:\sqrt[3]{637}}{7}$