can someone do a step by step explanation for me? I find this hard to understand...
also for questions like √(9 - 4√5) and √(4 + √15)
also for questions like √(9 - 4√5) and √(4 + √15)
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i) Let √(5 + 2√6) = √x + √y
ii) Squaring both sides, 5 + 2√6 = x + y + 2√(xy)
iii) Equating the rational and irrational parts on both sides,
x + y = 5 and xy = 6
iv) Applying, (x - y) = √{(x + y)² - 4xy}
(x - y) = 1
v) Thus solving (x + y) = 5 and (x - y) = 1, x = 3 and y = 2
So, √(5 + 2√6) = √3 + √2
Verify: Squaring right side,it is: 3 + 2√6 + 2 = 5 + 2√6.
Q.2:
i) In similar process as above, let √(9 - 4√5) = √x - √y [Here we adopt √x - √, since in the given one we have negative sign].
ii) Squaring and equating the rational and irrational parts.
x + y = 9 and xy = 20; since here 2√(xy) = 4√5.
As in preious here also we get (x - y) = 1
Solving, x = 5 and y = 4
Hence square root = √5 - 2
Q.3:
This is similar to the first one; you may solve yourself by keeping it as √x + √y.
Ans: √(5/2) + √(3/2)
ii) Squaring both sides, 5 + 2√6 = x + y + 2√(xy)
iii) Equating the rational and irrational parts on both sides,
x + y = 5 and xy = 6
iv) Applying, (x - y) = √{(x + y)² - 4xy}
(x - y) = 1
v) Thus solving (x + y) = 5 and (x - y) = 1, x = 3 and y = 2
So, √(5 + 2√6) = √3 + √2
Verify: Squaring right side,it is: 3 + 2√6 + 2 = 5 + 2√6.
Q.2:
i) In similar process as above, let √(9 - 4√5) = √x - √y [Here we adopt √x - √, since in the given one we have negative sign].
ii) Squaring and equating the rational and irrational parts.
x + y = 9 and xy = 20; since here 2√(xy) = 4√5.
As in preious here also we get (x - y) = 1
Solving, x = 5 and y = 4
Hence square root = √5 - 2
Q.3:
This is similar to the first one; you may solve yourself by keeping it as √x + √y.
Ans: √(5/2) + √(3/2)
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... √[ 5 + 2√6 ]
= √[ 3 + 2 √((3)(2)) ]
= √[ (√3)² + (√2)² + 2√(3).√(2) ]
= √[ (√3) + (√2) ]²
= (√3) + (√(2) ........................................ Ans.
___________________________
... 9 - 4√5
= 5 + 4 - 2(2)√5
= (√5)² + 2² - 2(2)(√5)
= [ (√5) - 2 ]²
√(9-4√5) = (√5) - 2 .................................. Ans.
____________________________
... 4 + √(15)
= (1/2)[ 8 + 2√(15) ]
= (1/2) [ 5 + 3 + 2√((5)(3)) ], ...
_____________________________
= √[ 3 + 2 √((3)(2)) ]
= √[ (√3)² + (√2)² + 2√(3).√(2) ]
= √[ (√3) + (√2) ]²
= (√3) + (√(2) ........................................ Ans.
___________________________
... 9 - 4√5
= 5 + 4 - 2(2)√5
= (√5)² + 2² - 2(2)(√5)
= [ (√5) - 2 ]²
√(9-4√5) = (√5) - 2 .................................. Ans.
____________________________
... 4 + √(15)
= (1/2)[ 8 + 2√(15) ]
= (1/2) [ 5 + 3 + 2√((5)(3)) ], ...
_____________________________
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I know of no way to simplify those. Any of those.