Question

√1+2008√1+2009√1+2010√1+2011.2013​

Answer 1

Answer:

√(1 + (2008(√1 + (2009(√1 + (2010√(1 + (2011.2013)))))))

= 2009

Step-by-step explanation:

We have,

√(1 + (2008(√1 + (2009(√1 + (2010√(1 + (2011.2013)))))))

We know that,

(x + y)(x - y) = x² - y²

Then,

We know that,

2011 × 2013 = (2012 - 1)(2012 + 1)

= (2012² - 1²)

So,

√(1 + (2008(√1 + (2009(√1 + (2010√(1 + (2011.2013)))))))

= √(1 + (2008(√1 + (2009(√1 + (2010√(1 + 2012² - 1²))))))

= √(1 + (2008(√1 + (2009(√1 + (2010√(1 + 2012² - 1))))))

= √(1 + (2008(√1 + (2009(√1 + (2010√(1 - 1 + 2012²))))))

= √(1 + (2008(√1 + (2009(√1 + (2010√(2012²))))))

= √(1 + (2008(√1 + (2009(√1 + (2010.2012)))))

Again Using the Identity,

= √(1 + (2008(√1 + (2009(√1 + (2011² - 1²)))))

= √(1 + (2008(√1 + (2009(√1 + (2011² - 1)))))

= √(1 + (2008(√1 + (2009(√1 - 1 + 2011²))))

= √(1 + (2008(√1 + (2009(√2011²))))

= √(1 + (2008(√1 + (2009.2011))))

Again Using the Identity,

= √(1 + (2008(√1 + (2010² - 1²))))

= √(1 + (2008(√1 + (2010² - 1))))

= √(1 + (2008(√1 - 1 + 2010²)))

= √(1 + (2008(√2010²)))

= √(1 + (2008.2010))

Again Using the Identity,

= √(1 + (2009² - 1²))

= √(1 + (2009² - 1))

= √(1 - 1 + 2009²)

= √2009²

= 2009

Hence,

√(1 + (2008(√1 + (2009(√1 + (2010√(1 + (2011.2013)))))))

= 2009