Question

Simplify : [1^3+2^3+3^3+4^3+5^3]^1/2

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Answer 1

$Given \: \: Question \: Is \: \: \\ \\ (1 {}^{3} + 2 {}^{3} + 3 {}^{3} + 4 {}^{3} + 5 {}^{3} ) {}^{ \frac{1}{2} } \\ \\ Answer \: \\ \\ we \: know \: that \: sum \: of \: cubes \: of \: natural \: \\ numbers \: is \: given \: by \: \frac{(n(n + 1)) {}^{2} }{4} \\ where \: n \: is \: number \: of \: terms. \\ \\ here \: n \: = 5 \\ \\ so \: sum \: of \: this \: series \: is \: given \: by \: \\ \\ ( \frac{(5(5 + 1)) {}^{2} }{4} ) {}^{ \frac{1}{2} } \\ \\ ( \frac{(30) {}^{2} }{4} ) {}^{ \frac{1}{2} } \\ \\ \frac{((30) {}^{2} ) {}^{ \frac{1}{2} } }{4 {}^{ \frac{1}{2} } } \\ \\ \frac{30}{(2 {}^{2} ) {}^{ \frac{1}{2} } } \\ \\ \frac{30}{2} \\ \\ = 15$

Answer 2

[1^3+2^3+3^3+4^3+5^3]^1/2

= [1+8+27+64+125]^1/2

= [225]^1/2

= 15

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