How do you calculate #sqrt(71 * 70 * 69 * 68 + 1)# ?

4 Answers
George C.
Jul 21, 2017

#4829#

Explanation:

If we put #x=139/2# then we have:

#sqrt(71*70*69*68+1) = sqrt((x+3/2)(x+1/2)(x-1/2)(x-3/2)+1)#

#color(white)(sqrt(71*70*69*68+1)) = sqrt((x^2-9/4)(x^2-1/4)+1)#

#color(white)(sqrt(71*70*69*68+1)) = sqrt(x^4-10/4x^2+9/16+1)#

#color(white)(sqrt(71*70*69*68+1)) = sqrt(x^4-10/4x^2+25/16)#

#color(white)(sqrt(71*70*69*68+1)) = sqrt((x^2-5/4)^2)#

#color(white)(sqrt(71*70*69*68+1)) = x^2-5/4#

#color(white)(sqrt(71*70*69*68+1)) = 139^2/4-5/4#

#color(white)(sqrt(71*70*69*68+1)) = (19321-5)/4#

#color(white)(sqrt(71*70*69*68+1)) = 19316/4#

#color(white)(sqrt(71*70*69*68+1)) = 4829#

John D.
Jul 21, 2017

#4829#

Explanation:

Remember the formula:

#(a-b)(a+b) = a^2 - b^2#

Using this, we can say that:

#70*69 = (69.5+1/2)(69.5-1/2) = 69.5^2 - (1/2)^2 = 69.5^2 - 1/4#

#71*68 = (69.5+3/2)(69.5-3/2) = 69.5^2 - (3/2)^2 = 69.5^2 - 9/4#

Therefore:

#sqrt(71*70*69*68+1)#

#= sqrt((69.5^2-1/4)(69.5^2-9/4) + 1#

Using the foil method:

#= sqrt(69.5^4 - 5/2 * 69.5^2 + 9/16 + 1)#

#= sqrt(69.5^4 - 5/2 * 69.5^2 + 25/16)#

Multiply by #sqrt16/4#:

#= sqrt(16(69.5^4 - 5/2 * 69.5^2 + 25/16))/4#

#= sqrt(2^4*69.5^4 - 4 * 5/2 * 2^2 * 69.5^2 + 16*25/16)/4#

Notice how I split up the 16 in the middle term so I can multiply part of it by the 5/2 and the rest of it by the 69.5^2.

#= sqrt(139^4 - 10 * 139^2 + 25)/4#

Replace #139^2# with #x# so we get a quadratic.

#= sqrt(x^2 - 10x + 25)/4#

#= sqrt((x-5)^2)/4#

#= (x-5)/4#

#= (139^2 - 5)/4#

Using long multiplication, we get:

#= (19321 - 5)/4#

#= 19316/4#

And finally using long division, we get:

#4829#

Final Answer

Barney V.
Jul 21, 2017

#+-4829#

Explanation:

#sqrt(71*70*69*68+1)#

#:.=sqrt(23319241)#

#:.=4829#

George C.
Jul 21, 2017

#sqrt(71*70*69*68+1) = 4829#

Explanation:

An easier way...

#70*70 = 4900#

So:

#70*69 = 4900-70 = 4830#

#70*68 = 4830-70 = 4760#

#71*68 = 4760+68 = 4828#

So:

#sqrt(71*70*69*68+1) = sqrt(4830*4828+1)#

#color(white)(sqrt(71*70*69*68+1)) = sqrt((4829+1)(4829-1)+1)#

#color(white)(sqrt(71*70*69*68+1)) = sqrt(4829^2-1^2+1)#

#color(white)(sqrt(71*70*69*68+1)) = sqrt(4829^2)#

#color(white)(sqrt(71*70*69*68+1)) = 4829#

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