Question

proove this
$( {1}^{3} + {2}^{3} + {3}^{3} + {4}^{3} + {5}^{3} )^{ \frac{1}{2} } + ( {5}^{3})^{ \frac{1}{3} }$
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Answer 1

Step-by-step explanation:

Proof :-

On taking LHS

(1³+2³+3³+4³+5³)½

We know that

The sum of the cubes of first n natural numbers

= [n(n+1)/2]²

We have,

n = 5

So ,

1³+2³+3³+4³+5³

=> [5(5+1)/2]²

=> [(5×6)/2]²

=> (30/2)²

=> (15)²

Now,

(1³+2³+3³+4³+5³)½

=> [(15)²]½

=> 15

Therefore, LHS = 15 -----------(1)

On taking RHS

(1³+2³+3³+4³)½ + (5³)⅓

We know that

The sum of the cubes of first n natural numbers

= [n(n+1)/2]²

We have,

n = 4

So ,

1³+2³+3³+4³+5³

=> [4(4+1)/2]²

=> [(4×5)/2]²

=> (20/2)²

=> (10)²

Now,

(1³+2³+3³+4³)½

=> [(10)²]½

=> 10

and

(5³)⅓ = 5

Now,

(1³+2³+3³+4³)½ + (5³)⅓

=> 10+5

=> 15

Therefore, RHS = 15 -----------(2)

From (1)&(2)

LHS = RHS

(1³+2³+3³+4³+5³)½ = (1³+2³+3³+4³)½ + (5³)⅓

Hance, Proved.

Used formulae:-

→ The sum of the cubes of first n natural numbers

= [n(n+1)/2]²