Square root problem

[Guest]

Can someone tell me how to do this without using a calculator.

Calculate: [tex]\sqrt{(1998)(1996)(1994)(1992)+16}[/tex]

[Guest]

Use the difference of two squares formula:
[tex]a^2-b^2 = (a-b)(a+b)[/tex]

[tex]\sqrt{1998\times 1996\times 1994\times 1992+16}[/tex]
[tex]= \sqrt{(1998\times 1992)\times 1996\times 1994+16}\;[/tex] (Apply difference of two squares to [tex]1998[/tex] and [tex]1992[/tex])
[tex]= \sqrt{(1995^2-3^2)\times 1996\times 1994+16}\;[/tex] (Apply difference of two squares to [tex]1996[/tex] and [tex]1994[/tex])
[tex]= \sqrt{(1995^2-3^2)\times (1995^2-1^2)+16}\;[/tex] (Expand the brackets)
[tex]= \sqrt{1995^4-10\times 1995^2+25}\;[/tex] (Factorize)
[tex]= \sqrt{(1995^2-5)^2}[/tex]
[tex]= \pm(1995^2-5)\;[/tex] (Rewrite [tex]1995[/tex])
[tex]= \pm((2000-5)^2-5)\;[/tex] (Expand the brackets)
[tex]= \pm(2000^2-10\times 2000+20)[/tex]
[tex]= \pm 3980020[/tex]

Hope this helped,

R. Baber.

[Guest]

Wow, thanks very much. I wrote it all down and now I'll try to follow it through.