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Maths-

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Question

Simplify $fraction numerator 1 over denominator 2 plus square root of 5 end fraction plus fraction numerator 1 over denominator square root of 5 plus square root of 6 end fraction plus fraction numerator 1 over denominator square root of 6 plus square root of 7 end fraction plus fraction numerator 1 over denominator square root of 7 plus square root of 8 end fraction$

hint Hint:

Hint:
Rationalization is a technique which is used to eliminate the irrational number in the denominator. For example, $fraction numerator 1 over denominator square root of 3 end fraction$
can be rationalised as $fraction numerator 1 over denominator square root of 3 end fraction cross times fraction numerator square root of 3 over denominator square root of 3 end fraction equals fraction numerator square root of 3 over denominator 3 end fraction$

The correct answer is: 2 sqrt2-2

Let’s say x = $fraction numerator 1 over denominator 2 plus square root of 5 end fraction plus fraction numerator 1 over denominator square root of 5 plus square root of 6 end fraction plus fraction numerator 1 over denominator square root of 6 plus square root of 7 end fraction plus fraction numerator 1 over denominator square root of 7 plus square root of 8 end fraction horizontal ellipsis horizontal ellipsis text (1) end text$
Step 1 of 2:
Let’s rationalise every term one by one:
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell fraction numerator 1 over denominator 2 plus square root of 5 end fraction equals fraction numerator 1 over denominator 2 plus square root of 5 end fraction cross times fraction numerator 2 minus square root of 5 over denominator 2 minus square root of 5 end fraction equals fraction numerator 2 minus square root of 5 over denominator 4 minus 5 end fraction equals square root of 5 minus 2 end cell row cell fraction numerator 1 over denominator square root of 5 plus square root of 6 end fraction equals fraction numerator 1 over denominator square root of 5 plus square root of 6 end fraction cross times fraction numerator square root of 5 minus square root of 6 over denominator square root of 5 minus square root of 6 end fraction equals fraction numerator square root of 5 minus square root of 6 over denominator 5 minus 6 end fraction equals square root of 6 minus square root of 5 end cell row cell fraction numerator 1 over denominator square root of 6 plus square root of 7 end fraction equals fraction numerator 1 over denominator square root of 6 plus square root of 7 end fraction cross times fraction numerator square root of 6 minus square root of 7 over denominator square root of 6 minus square root of 7 end fraction equals fraction numerator square root of 6 minus square root of 7 over denominator 6 minus 7 end fraction equals square root of 7 minus square root of 6 end cell row cell fraction numerator 1 over denominator square root of 7 plus square root of 8 end fraction equals fraction numerator 1 over denominator square root of 7 plus square root of 8 end fraction cross times fraction numerator square root of 7 minus square root of 8 over denominator square root of 7 minus square root of 8 end fraction equals fraction numerator square root of 7 minus square root of 8 over denominator 7 minus 8 end fraction equals square root of 8 minus square root of 7 end cell end table$
Step 1 of 2:
Put all the rationalised value in equation (1)
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell x equals square root of 5 minus 2 plus square root of 6 minus square root of 5 plus square root of 7 minus square root of 6 plus square root of 8 minus square root of 7 end cell row cell x equals negative 2 plus square root of 8 end cell row cell x equals 2 square root of 2 minus 2 end cell end table$

Final Answer:
Hence, the simplified form of $fraction numerator 1 over denominator 2 plus square root of 5 end fraction plus fraction numerator 1 over denominator square root of 5 plus square root of 6 end fraction plus fraction numerator 1 over denominator square root of 6 plus square root of 7 end fraction plus fraction numerator 1 over denominator square root of 7 plus square root of 8 end fraction$ is $2 square root of 2 minus 2$

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