State whether the following statement is true or False

$\sqrt[2]{15}$ + $\sqrt[2]{3}$ > $\sqrt[2]{10}$ + $\sqrt[2]{8}$

True

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False

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Solution

The correct option is B
False

Let X = $\sqrt[2]{15}$ + $\sqrt[2]{3}$

Y = $\sqrt[2]{10}$ + $\sqrt[2]{8}$

$X^{2}$ = $(\sqrt[2]{15} + \sqrt[2]{3})^{2}$ = $(\sqrt[2]{15})^{2}$ + $(\sqrt[2]{3})^{2}$ + 2 $\sqrt[2]{45}$ = 18 + 2 $\sqrt[2]{45}$

$Y^{2}$ = $(\sqrt[2]{10} + \sqrt[2]{8})^{2}$ = $(\sqrt[2]{10})^{2}$ + $(\sqrt[2]{8})^{2}$ + 2 $\sqrt[2]{80}$ = 18 + 2 $\sqrt[2]{80}$

So clearly we can see 18 + 2 $\sqrt[2]{45}$ < 18 + 2 $\sqrt[2]{80}$ as 2 $\sqrt[2]{45}$ < 2 $\sqrt[2]{80}$

which implies $X^{2}$ < $Y^{2}$ and X < Y

hence the statement is false

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