Question

Evaluate: 6³+7³+8³+9³+10³ using property

Answer 1

Step-by-step explanation:

${(6 + 7 + 8 + 9 + 10)}^{3} \\ \\ = {40}^{3} \\ \\ = 40 \times 40 \times 40 \\ \\ = 1600 \times 40 \\ \\ = 64000$

Answer 2

Answer:

Required value of 6³+7³+8³+9³+10³ is 2800.

Step-by-step explanation:

Given,

${6}^{3} + {7}^{3} + {8}^{3} + {9}^{3} + {10}^{3}$

We want to find value of this expression.

We can write,

${6}^{3} + {7}^{3} + {8}^{3} + {9}^{3} + {10}^{3} = ( {1}^{3} + {2}^{3} + {3}^{3} + {4}^{3} + {5}^{3} ) + {6}^{3} + {7}^{3} + {8}^{3} + {9}^{3} + {10}^{3} - ( {1}^{3} + {2}^{3} + {3}^{3} + {4}^{3} + {5}^{3} )$

We know sum of cube of n natural numbers

$= \frac{ {n}^{2} {(n + 1)}^{2} }{4}$

Or,

${1}^{3} + {2}^{3} + {3}^{3} + ...... {n}^{3} = \frac{ {n}^{2} {(n + 1)}^{2} }{4}$

Now,

${1}^{3} + {2}^{3} + {3}^{3} + {4}^{3} + {5}^{3} = \frac{ {5}^{2} {(5 + 1)}^{2} }{4} \\ = \frac{25 \times 36}{4} \\ = 25 \times 9 \\ = 225$

and

${1}^{3} + {2}^{3} + {3}^{3} + {4}^{3} + {5}^{3} + ....... + {10}^{3} = \frac{ {10}^{2} {(10 + 1)}^{2} }{4}$

$= \frac{100 \times 121}{4} \\ = 25 \times 121 \\ = 3025$

So,

${6}^{3} + {7}^{3} + {8}^{3} + {9}^{3} + {10}^{3} \\ = 3025 - 225 \\ = 2800$

Therefore required value of 6³+7³+8³+9³+10³ is 2800.

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