Answer:
[tex]\left(2\:\sqrt{3\:+\:3\:\sqrt{2}\:\:}\:\right)\:^2=4(3+3\sqrt{2})[/tex]
Step-by-step explanation:
Considering the radical expression
[tex]\left(2\:\sqrt{3\:+\:3\:\sqrt{2}\:\:}\:\right)\:^2[/tex]
[tex]\mathrm{Apply\:exponent\:rule}:\quad \left(a\cdot \:b\right)^n=a^nb^n[/tex]
[tex]=2^2\left(\sqrt{3+3\sqrt{2}}\right)^2.....[A][/tex]
Simplifying
[tex]\left(\sqrt{3+3\sqrt{2}}\right)^2[/tex]
[tex]\mathrm{Apply\:radical\:rule}:\quad \sqrt{a}=a^{\frac{1}{2}}[/tex]
[tex]=\left(\left(3+3\sqrt{2}\right)^{\frac{1}{2}}\right)^2[/tex]
[tex]\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}[/tex]
[tex]=\left(3+3\sqrt{2}\right)^{\frac{1}{2}\cdot \:2}[/tex]
[tex]=3+3\sqrt{2}[/tex] As [tex]\frac{1}{2}\cdot \:2=1[/tex]
So, putting [tex]3+3\sqrt{2}[/tex] into Equation [A]
[tex]=2^2\left(\sqrt{3+3\sqrt{2}}\right)^2.....[A][/tex]
As
[tex]\left(\sqrt{3+3\sqrt{2}}\right)^2=3+3\sqrt{2}[/tex]
So, Equation [A] becomes
[tex]=2^{2}(3+3\sqrt{2})[/tex]
[tex]=4(3+3\sqrt{2})[/tex]
Therefore,
[tex]\left(2\:\sqrt{3\:+\:3\:\sqrt{2}\:\:}\:\right)\:^2=4(3+3\sqrt{2})[/tex]
Keywords: radical expression
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