Multiplication of radicals

Example 1: Multiply and simplify $\sqrt{15}\:\sqrt{3}$

Solution:

$\begin{align*} \sqrt{15}\:\sqrt{3}&=\sqrt{15\cdot3}\\ &=\sqrt{3\cdot 5\cdot 3}\\ &=3\sqrt{5}\\ \end{align*}$

Example 2: Multiply and simplify $\sqrt{24u^3y^6}\:\sqrt{32u^5y^7}$

Solution:

$\begin{align*} \sqrt{24u^3y^6}\:\sqrt{32u^5y^7}&=\sqrt{2^3\cdot 3u^3y^6}\:\sqrt{2^5u^5y^7}\\ &=\sqrt{2^3\cdot 3\cdot 2^5u^8y^{13}}\\ &=\sqrt{2^8\cdot 3u^8y^{12}y}\\ &=2^4u^4y^6\sqrt{3y}\\ &=16u^4y^6\sqrt{3y}\\ \end{align*}$

Example 3: Multiply and simplify $\sqrt[4]{18xy^3}\:\sqrt[4]{27x^5y}$

Solution:

$\begin{align*} \sqrt[4]{18xy^3}\:\sqrt[4]{27x^5y}&=\sqrt[4]{18\cdot 27 x^6y^4}\\ &=\sqrt[4]{2\cdot 3^2\cdot 3^3 x^6y^4}\\ &=\sqrt[4]{2\cdot 3^4\cdot 3 x^4x^2y^4}\\ &=3xy\sqrt[4]{2\cdot 3 x^2}\\ &=3xy\sqrt[4]{6 x^2}\\ \end{align*}$

Example 4: Multiply, $11\sqrt{7}(\sqrt{7}-2\sqrt{6})$

Solution:

$\begin{align*} 11\sqrt{7}(\sqrt{7}-2\sqrt{6})&=11\sqrt{7}\cdot \sqrt{7}-11\sqrt{7}\cdot 2\sqrt{6}\\ &=11\cdot 7-22\sqrt{7\cdot 6}\\ &=77-22\sqrt{42}\\ \end{align*}$

Example 5: Multiply, $(5\sqrt{3}+4\sqrt{11})(6\sqrt{15}-3\sqrt{6})$

Solution:

$\begin{align*} (5\sqrt{3}+4\sqrt{11})(6\sqrt{15}-3\sqrt{6})&=5\sqrt{3}\cdot 6\sqrt{15}-5\sqrt{3}\cdot 3\sqrt{6}+4\sqrt{11}\cdot 6\sqrt{15}- 4\sqrt{11}\cdot 3\sqrt{6}\\ &=30\sqrt{3\cdot 15}-15\sqrt{3\cdot 6}+24\sqrt{11\cdot 15}- 12\sqrt{11\cdot 6}\\ &=90\sqrt{5}-15\sqrt{3\cdot 6}+24\sqrt{11\cdot 15}- 12\sqrt{11\cdot 6}\\ &=90\sqrt{5}-45\sqrt{2}+24\sqrt{165}- 12\sqrt{66}\\ \end{align*}$

Example 6: Multiply, $(3\,\sqrt{a}+\sqrt{15})(3\, \sqrt{a}-\sqrt{15})$

Solution:

The expression is in the form $(a+b)(a-b)$, so we use the formula $(a+b)(a-b)=a^2-b^2$.

$\begin{align*} (3\sqrt{a}+\sqrt{15})(3\sqrt{a}-\sqrt{15})&=(3\sqrt{a})^2-(\sqrt{15})^2\\ &=9a-15\\ \end{align*}$

Example 7: Multiply, $(5\sqrt{y}+\sqrt{6})^2$

Solution:

Use the formula $(a+b)^2=a^2+2ab+b^2$, or use $(a+b)^2=(a+b)(a+b)$.

$\begin{align*} (5\sqrt{y}+\sqrt{6})^2&=(5\sqrt{y}+\sqrt{6})\cdot (5\sqrt{y}+\sqrt{6})\\ &=5\sqrt{y} \cdot 5\sqrt{y}+5\sqrt{y} \cdot \sqrt{6}+\sqrt{6}\cdot 5\sqrt{y}+\sqrt{6}\cdot \sqrt{6}\\ &=25y+5\sqrt{6y}+5\sqrt{6y}+6\\ &=25y+10\sqrt{6y}+6\\ \end{align*}$

Example 8: Simplify, $\sqrt[8]{x^5}\:\sqrt[3]{x}$

Solution:

$\begin{align*} \sqrt[8]{x^5}\:\sqrt[3]{x}&=x^{5/8}\cdot x^{1/3}\\ &=x^{5/8+1/3}\\ &=x^{23/24}\\ &=\sqrt[24]{x^{23}}\\ \end{align*}$

Example 9: Simplify, $\dfrac{\sqrt[7]{u^6}}{\sqrt[5]{u^2}}$

Solution:

$\begin{align*} \dfrac{\sqrt[7]{u^6}}{\sqrt[5]{u^2}} &= \dfrac{u^{6/7}}{u^{2/5}}\\ &= u^{6/7-2/5}\\ &=u^{16/35}\\ &=\sqrt[35]{u^{16}}\\ \end{align*}$