Question

Match each expression to its equivalent expression with a rational denominator.

Answer 1

\dfrac{1}{\sqrt[4]{3x^3y^5}}\cdot\dfrac{\sqrt[4]{3^3xy^3}}{\sqrt[4]{3^3xy^3}}=\dfrac{\sqrt[4]{3^3xy^3}}{\sqrt[4]{3^4x^4y^8}}=\dfrac{\sqrt[4]{3^3xy^3}}{{\sqrt[4]{3^4x^4(y^2)^4}}}=\dfrac{\sqrt[4]{3^3xy^3}}{3xy^2}

\dfrac{3}{\sqrt[4]{27x^{11}y^{13}}}=\dfrac{3}{\sqrt[4]{3^3x^{11}y^{13}}}=\dfrac{3}{\sqrt[4]{27x^{11}y^{13}}}\cdot\dfrac{\sqrt[4]{3xy^3}}{\sqrt[4]{3xy^3}}=\dfrac{3\sqrt[4]{3xy^3}}{\sqrt[4]{3^4x^{12}y^{16}}}\\\\=\dfrac{3\sqrt[4]{3xy^3}}{\sqrt[4]{3^4(x^3)^4(y^4)^4}}=\dfrac{3\sqrt[4]{3xy^3}}{3x^3y^4}=\dfrac{\sqrt[4]{3xy^3}}{x^3y^4}

\dfrac{2}{\sqrt[6]{2x^7y^5}}=\dfrac{2}{\sqrt[6]{2x^7y^5}}\cdot\dfrac{\sqrt[6]{2^5x^5y}}{\sqrt[6]{2^5x^5y}}=\dfrac{2\sqrt[6]{2^5x^5y}}{\sqrt[6]{2^6x^{12}y^6}}=\dfrac{2\sqrt[6]{2^5x^5y}}{\sqrt[6]{2^6(x^2)^6y^6}}\\\\=\dfrac{2\sqrt[6]{2^5x^5y}}{2x^2y}=\dfrac{\sqrt[6]{32x^5y}}{x^2y}

\dfrac{4}{\sqrt[6]{32x^5y^9}}=\dfrac{4}{\sqrt[6]{2^5x^5y^9}}\cdot\dfrac{\sqrt[6]{2xy^3}}{\sqrt[6]{2xy^3}}=\dfrac{4\sqrt[6]{2xy^3}}{\sqrt[6]{2^6x^6y^{12}}}=\dfrac{4\sqrt[6]{2xy^3}}{\sqrt[6]{2^6x^6(y^2)^6}}\\\\=\dfrac{4\sqrt[6]{2xy^3}}{2xy^2}=\dfrac{2\sqrt[6]{2xy^3}}{xy^2}}

\text{Used:}\\\\\sqrt[n]{a^n}=a\\\\(a^n)^m=a^{n\cdot m}\\\\a^n\cdot a^m=a^{n+m}\\\\\sqrt[n]{a\cdot b}=\sqrt[n]a\cdot\sqrt[n]b

Answer 2

$\dfrac{1}{\sqrt[4]{3x^3y^5}}\cdot\dfrac{\sqrt[4]{3^3xy^3}}{\sqrt[4]{3^3xy^3}}=\dfrac{\sqrt[4]{3^3xy^3}}{\sqrt[4]{3^4x^4y^8}}=\dfrac{\sqrt[4]{3^3xy^3}}{{\sqrt[4]{3^4x^4(y^2)^4}}}=\dfrac{\sqrt[4]{3^3xy^3}}{3xy^2}$

$\dfrac{3}{\sqrt[4]{27x^{11}y^{13}}}=\dfrac{3}{\sqrt[4]{3^3x^{11}y^{13}}}=\dfrac{3}{\sqrt[4]{27x^{11}y^{13}}}\cdot\dfrac{\sqrt[4]{3xy^3}}{\sqrt[4]{3xy^3}}=\dfrac{3\sqrt[4]{3xy^3}}{\sqrt[4]{3^4x^{12}y^{16}}}\\\\=\dfrac{3\sqrt[4]{3xy^3}}{\sqrt[4]{3^4(x^3)^4(y^4)^4}}=\dfrac{3\sqrt[4]{3xy^3}}{3x^3y^4}=\dfrac{\sqrt[4]{3xy^3}}{x^3y^4}$

$\dfrac{2}{\sqrt[6]{2x^7y^5}}=\dfrac{2}{\sqrt[6]{2x^7y^5}}\cdot\dfrac{\sqrt[6]{2^5x^5y}}{\sqrt[6]{2^5x^5y}}=\dfrac{2\sqrt[6]{2^5x^5y}}{\sqrt[6]{2^6x^{12}y^6}}=\dfrac{2\sqrt[6]{2^5x^5y}}{\sqrt[6]{2^6(x^2)^6y^6}}\\\\=\dfrac{2\sqrt[6]{2^5x^5y}}{2x^2y}=\dfrac{\sqrt[6]{32x^5y}}{x^2y}$

$\dfrac{4}{\sqrt[6]{32x^5y^9}}=\dfrac{4}{\sqrt[6]{2^5x^5y^9}}\cdot\dfrac{\sqrt[6]{2xy^3}}{\sqrt[6]{2xy^3}}=\dfrac{4\sqrt[6]{2xy^3}}{\sqrt[6]{2^6x^6y^{12}}}=\dfrac{4\sqrt[6]{2xy^3}}{\sqrt[6]{2^6x^6(y^2)^6}}\\\\=\dfrac{4\sqrt[6]{2xy^3}}{2xy^2}=\dfrac{2\sqrt[6]{2xy^3}}{xy^2}}$

$\text{Used:}\\\\\sqrt[n]{a^n}=a\\\\(a^n)^m=a^{n\cdot m}\\\\a^n\cdot a^m=a^{n+m}\\\\\sqrt[n]{a\cdot b}=\sqrt[n]a\cdot\sqrt[n]b$