Question

Without calculator
sqrt(500*501*502*503+1)=?

Answer 1

Answer:

251501

Step-by-step explanation:

Given :

To find the value of $\sqrt{(500*501*502*503)+1}$

Without using calculator, (without direct multiplication)

Solution :

Note:

a² - b² = ( a + b ) ( a - b )   ...(i)

(a + b)² = a² + 2ab + b²     ...(ii)

(a - b)² = a² - 2ab + b²       ...(iii)

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$\sqrt{(500*501*502*503)+1}$

$\sqrt{(500*502*501*503)+1}$

$\sqrt{((501 -1)*(501+1)*(502-1)*(502+1))+1}$

$\sqrt{((501^2 -1)*(502^2-1))+1}$   by (i)

$\sqrt{((501^2*502^2 - 501^2 - 502^2 + 1)+1}$

$\sqrt{(501^2*(501+1)^2 - 501^2 - (501+1)^2 + 2}$

$\sqrt{(501^2*[(501)^2 + 2*501 + 1)] - 501^2 - [(501)^2 + 2*501 +1] + 2}$

$\sqrt{(501^2*[(501)^2 + 2*501 + 1 - 1] - [(501)^2 + 2*501 +1] + 2}$

$\sqrt{(501^2*[(501)^2 + 2*501] - (501)^2 - 2*501 - 1 + 2}$

$\sqrt{(501^4 + 2*501^3 - (501)^2 - 2*501 + 1}$

$\sqrt{(501^4 + 2*501^3 - 2*501 - (501^2 - 1) }$

$\sqrt{(501^4 + 2*501(501^2 - 1) - (501^2 - 1) }$

$\sqrt{(501^4 + (2*501 - 1)(501^2 - 1)}$

$\sqrt{(501^4 + (2*(500+1) - 1)(501^2 - 1)}$

$\sqrt{(501^4 + (2*(500) +2*1 - 1)(501^2 - 1)}$

$\sqrt{(501^4 + (2*(500)+2 - 1)(501^2 - 1)}$

$\sqrt{(501^4 + (2*(500+1)(501^2 - 1)}$

$\sqrt{(501^4 + (2*(500+1)(501^2 - 1)}$

$\sqrt{(501^4 + (2*(500)(501^2 - 1) + (501^2 -1)}$

$\sqrt{(501^4 + (2*(500)(501^2) -2(500) + (501^2 -1)}$

$\sqrt{(501^4 + (2*(500)(501^2) + (501^2 -2(500) -1)}$

$\sqrt{(501^4 + (2*(500)(501^2) + (501^2 -2(500 +1 -1) -1 )}$

$\sqrt{(501^4* + (2*(500)(501^2) + (501^2 -2(500 +1 )- 2*(-1) -1 )}$

$\sqrt{(501^4* + (2*(500)(501^2) + (501^2 -2(501) + 2 -1 )}$

$\sqrt{(501^4* + (2*(500)(501^2) + (501^2 -2(501) + 1 )}$

$\sqrt{(501^4* + (2*(500)(501^2) + (501 - 1)^2}$ by (iii)

$\sqrt{(501^4* + (2*(500)(501^2) + (500)^2}$

$\sqrt{(501^2 + 500)^2}$ by (ii)

$(501^2 + 500) = ((500)^2 + 2*500 + 1 + 500) = 250000 + 1000 + 500 + 1 = 251501$

Answer 2

Answer:

Here is the Solution :-

Step-by-step explanation:

Given in the image beside,

Easy & Crisp way.