Without resorting to the use of a calculator or computer, find a simpler representation for each of the numbers below: sqrt (2+sqrt 3)-sqrt (2-sqrt 3)

armlanna1sK

armlanna1sK

Answered question

2022-11-25

Without resorting to the use of a calculator or computer, find a simpler representation for each of the numbers below:
2 + 3 2 3
Extend to find:
2 + 3 + 2 3
Generalise to find: a + b ± a b

Answer & Explanation

Maryjane Estrada

Maryjane Estrada

Beginner2022-11-26Added 10 answers

Notice that 2 2 = 3 2 + 1 2 and
2 ( 3 ± 1 ) 2 2 = 3 ± 1 2 .
Hence your expressions are 6 and 2 .
To generalize, write
a ± b = ( α ± β b ) 2 = α 2 + β 2 b ± 2 α β b
and identify.

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