a 2 − b 2 a^{2} - b^{2}

Algebra Level 2

If a = 2 2005 + 2 − 2005 a = 2^{2005} + 2^{-2005} and b = 2 2005 − 2 − 2005 b = 2^{2005} - 2^{-2005} then what is a 2 − b 2 a^{2} - b^{2} ?

4 2 3 0 1

Rafik Hawking by Rafik Hawking , 18, Belgium

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2 solutions

Arjen Vreugdenhil
Nov 13, 2017

In general, if a = x + y a = x + y and b = x − y b = x - y , then a 2 − b 2 = ( x + y ) 2 − ( x − y ) 2 = ( x 2 + 2 x y + y 2 ) − ( x 2 − 2 x y + y 2 ) = 2 x y − ( − 2 x y ) = 4 x y . a^2 - b^2 = (x+y)^2 - (x-y)^2 = (x^2+2xy +y^2) - (x^2-2xy+y^2) = 2xy - (-2xy) = 4xy. In this case, a 2 − b 2 = 4 ⋅ 2 2005 ⋅ 2 − 2005 = 4 ⋅ 1 = 4. a^2 - b^2 = 4\cdot 2^{2005}\cdot 2^{-2005} = 4\cdot 1 = 4.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

good explanation

Rafik Hawking - 3 years, 6 months ago
Kb E
Nov 13, 2017

a 2 − b 2 = ( a − b ) ( a + b ) = 2 ⋅ 2 − 2005 ⋅ 2 ⋅ 2 2005 = 4 a^2-b^2 = (a-b)(a+b) = 2\cdot 2^{-2005}\cdot 2 \cdot 2^{2005} = 4 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

good explanation

Rafik Hawking - 3 years, 6 months ago

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