Operaciones con raíces. Radicales (2)

a) \(\displaystyle\left(\sqrt{3}\sqrt{x}\right)^2=\sqrt{3x}^2=3x\)

b)  \(\displaystyle\left(5\sqrt{3x}\right)^3=5^3\sqrt{(3x)^3}=5^3\sqrt{3^3x^3}=5^3\cdot3x\sqrt{3x}=375x\sqrt{3x}\)

c)  \(\displaystyle\left(\sqrt{2}-\sqrt{3}\right)^2=\sqrt{2}^2-2\sqrt{2}\sqrt{3}+\sqrt{3}^2=2-2\sqrt{6}+3=5-2\sqrt{6}\)

d)  \(\displaystyle\left(\sqrt{x}+\sqrt{3}\right)^2=\sqrt{x}^2+2\sqrt{x}\sqrt{3}+\sqrt{3}^2=x+2\sqrt{3x}+3\)

e)  \(\displaystyle\left(\sqrt{x+y}-\sqrt{x-y}\right)\left(\sqrt{x+y}+\sqrt{x-y}\right)=\)

\(=\left(\sqrt{x+y}\right)^2-\left(\sqrt{x-y}\right)^2=x+y-(x-y)=x+y-x+y=2y\)

f)  \(\displaystyle\left(\sqrt{5x}+\sqrt{2x}\right)\left(\sqrt{5x}-\sqrt{2x}\right)=\sqrt{5x}^2-\sqrt{2x}^2=5x-2x=3x\)

a)  \(\displaystyle\sqrt{36a^2x^3y^4z^5}=\sqrt{2^23^2a^2x^3y^4z^5}=2\cdot3axy^2z^2\sqrt{xz}=\)

\(=6axy^2z^2\sqrt{xz}\)

b)  \(\displaystyle\sqrt[3]{24x^7y^5}=\sqrt[3]{2^33x^7y^5}=2x^2y\sqrt[3]{3xy^2}\)

c)  \(\displaystyle\sqrt[4]{\frac{a^5b^6z^7}{64}}=\sqrt[4]{\frac{a^5b^6z^7}{2^6}}=\frac{abc}{2}\sqrt[4]{\frac{ab^2z^3}{2^2}}\)

d)  \(\displaystyle\sqrt[3]{\frac{54x^3}{a^3b^6}}=\sqrt[3]{\frac{2\cdot3^3x^3}{a^3b^6}}=\frac{3x}{ab^2}\sqrt[3]{2}\)

a)   \(\displaystyle2x\sqrt{\frac{1}{x}}=\sqrt{\frac{(2x)^2}{x}}=\sqrt{\frac{4x^2}{x}}=\sqrt{4x}\)

b)  \(\displaystyle2\sqrt[3]{\frac{1}{16}}=\sqrt[3]{\frac{2^3}{16}}=\sqrt[3]{\frac{2^3}{2^4}}=\sqrt[3]{\frac{1}{2}}\)

c)  \(\displaystyle a^2xy\sqrt[3]{ax^2y^2}=\sqrt[3]{(a^2)^3x^3y^3ax^2y^2}=\sqrt[3]{a^7x^5y^5}\)

d)  \(\displaystyle6x^2y\sqrt{\frac{y}{6x}}=\sqrt{\frac{6^2x^4y^2y}{6x}}=\sqrt{6x^3y^3}\)

a) \(\sqrt{2x}\cdot\sqrt{2xy}\cdot\sqrt{2xyz}\cdot\sqrt{2xy^2z^3}=\sqrt{2^4x^4y^4z^4}=\)

\(=2^2x^2y^2z^2=4x^2y^2z^2\)

b)  \(\displaystyle\sqrt{3}\cdot\sqrt[3]{9}\cdot\sqrt[4]{18}=\sqrt[12]{3^6}\sqrt[12]{(3^2)^4}\sqrt[12]{(3^22)^3}=\)

\(=\sqrt[12]{3^63^83^62^3}=\sqrt[12]{3^{20}2^3}=3\sqrt[12]{3^82^3}\)

c)  \(\displaystyle\sqrt{2x}\cdot\sqrt[4]{4x^3}\cdot\sqrt[6]{8x^5}=\sqrt[12]{2^6x^6}\sqrt[12]{(2^2)^3(x^3)^3}\sqrt[12]{(2^3)^2(x^5)^2}=\)

\(\displaystyle=\sqrt[12]{2^{18}x^{25}}=2x^2\sqrt[12]{2^6x}\)

d)  \(\displaystyle\sqrt{ab}\cdot\sqrt[3]{a^2b}\cdot\sqrt[6]{a^5b^3}=\sqrt[6]{a^3b^3}\sqrt[6]{(a^2)^2b^2}\sqrt[6]{a^5b^3}=\sqrt[6]{a^{12}b^8}=\)

\(\displaystyle=a^2b\sqrt[6]{b^2}=a^2b\sqrt[3]{b}\)

a)  \(\displaystyle6\sqrt{2}+\sqrt{50}-\sqrt{72}-\sqrt{32}=6\sqrt{2}+\sqrt{5^22}-\sqrt{3^22^3}-\sqrt{2^5}=\)

\(=\displaystyle6\sqrt{2}+5\sqrt{2}-3\cdot2\sqrt{2}-2^2\sqrt{2}=(6+5-6-4)\sqrt{2}=\sqrt{2}\)

b)  \(\displaystyle7\sqrt{63}+4\sqrt{28}-\sqrt{343}+\sqrt{7}=7\sqrt{3^27}+4\sqrt{2^27}-\sqrt{7^3}+\sqrt{7}=\)

\(\displaystyle=7\cdot3\sqrt{7}+4\cdot2\sqrt{7}-7\sqrt{7}+\sqrt{7}=(21+8-7+1)\sqrt{7}=23\sqrt{7}\)

a)  \(\displaystyle\frac{5}{\sqrt{5}}=\frac{5\sqrt{5}}{\sqrt{5}\sqrt{5}}=\frac{5\sqrt{5}}{5}=\sqrt{5}\)

b)  \(\displaystyle\frac{1}{\sqrt[3]{2}}=\frac{1\cdot\sqrt[3]{2^2}}{\sqrt[3]{2}\sqrt[3]{2^2}}=\frac{\sqrt[3]{4}}{\sqrt[3]{2^3}}=\frac{\sqrt[3]{4}}{2}\)

c)  \(\displaystyle\frac{2+\sqrt{3}}{2-\sqrt{3}}=\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}=\frac{2^2+2\cdot2\cdot\sqrt{3}+\sqrt{3}^2}{2^2-\sqrt{3}^2}=\)

\(\displaystyle=\frac{4+4\sqrt{3}+3}{4-3}=\frac{7+4\sqrt{3}}{1}=7+4\sqrt{3}\)

d)  \(\displaystyle\frac{5}{\sqrt{10}-\sqrt{5}}=\frac{5\left(\sqrt{10}+\sqrt{5}\right)}{\left(\sqrt{10}-\sqrt{5}\right)\left(\sqrt{10}+\sqrt{5}\right)}=\frac{5\left(\sqrt{10}+\sqrt{5}\right)}{\sqrt{10}^2-\sqrt{5}^2}=\)

\(\displaystyle=\frac{5\left(\sqrt{10}+\sqrt{5}\right)}{10-5}=\frac{5\left(\sqrt{10}+\sqrt{5}\right)}{5}=\sqrt{10}+\sqrt{5}\)