Simplification / 117
Ex. 7
Solve:
(835 + 378)2 + (835 - 378)2
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835 × 835 + 378 × 378
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835 × 835 + 378 × 378
We know that:
(a + b)2 + (a - b)2 = 2(a2 + b2)
(a + b)2 + (a - b)2 = 2(a2 + b2)
So,
(835 + 378)2 + (835 - 378)2 = 2(8352 + 3782)
= 2(8352 + 3782) / (8352 + 3782) = 2
Answer: 2
Ex. 8
If x = 3 + √8, then find the value of x2 + 1/x2.
x = 3 + √8
1/x = 1/(3 + √8)
Rationalizing denominator:
= (3 - √8) / [(3 + √8)(3 - √8)] = (3 - √8) / (9 - 8) = 3 - √8
Rationalizing denominator:
= (3 - √8) / [(3 + √8)(3 - √8)] = (3 - √8) / (9 - 8) = 3 - √8
x + 1/x = (3 + √8) + (3 - √8) = 6
(x + 1/x)2 = x2 + 1/x2 + 2
62 = x2 + 1/x2 + 2
36 = x2 + 1/x2 + 2
x2 + 1/x2 = 34
62 = x2 + 1/x2 + 2
36 = x2 + 1/x2 + 2
x2 + 1/x2 = 34
Answer: 34
Ex. 9
If a + 1/a = 4√2, then what is the value of a6 + a-6?
a + 1/a = 4√2
Squaring both sides:
(a + 1/a)2 = a2 + 2 + 1/a2 = (4√2)2 = 32
(a + 1/a)2 = a2 + 2 + 1/a2 = (4√2)2 = 32
a2 + 1/a2 = 32 - 2 = 30
Now, taking cube on both sides:
(a2 + 1/a2)3 = a6 + 1/a6 + 3(a2 + 1/a2)
(a2 + 1/a2)3 = a6 + 1/a6 + 3(a2 + 1/a2)
303 = a6 + 1/a6 + 3 × 30
27000 = a6 + 1/a6 + 90
27000 = a6 + 1/a6 + 90
a6 + 1/a6 = 27000 - 90 = 26910
Answer: 26910
Ex. 10
If x + 1/x = 2, then find the value of x2013 + 1/x2014.
x + 1/x = 2 ⇒ x = 1
x2013 + 1/x2014 = 1 + 1 = 2
Answer: 2