Given a + b = 9 and a – b = 2, what is a^4 + b^4?
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We have a + b = 9 and a – b= 2. The easiest way to find a^4 + b^4 would be to first use the given values of a + b and a – b to determine a and b.
a + b = 9 … (1)
a – b = 2 … (2)
(2) + (1)
=> a + b + a – b = 9 + 2
=> 2a = 11
=> a = 11/2
substituting this in (1)
=> 11/2 + b = 9
=> b = 9 – 11/2
=> b = (18 – 11) /2
=> b = 7/2
a^4 + b^4
=> (11/2)^4 + (7/2)^4
=> 14641 / 16 + 2401 / 16
=> 17042 / 16
=> 8521 / 8
Therefore a^4 + b^4 = 8521/8
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1st equation: a + b = 9
2nd equatrion: a - b = 2
since a+b=9, then a=9-b
replace this value of a to the 2nd equation.
since 2nd equation is a-b=2, then
(9-b)-b=2
9-b-b=2
9-2b=2
-2b=2-9
-2b=-7
b=7/2
given that b=7/2, substitute the value of b to one of the two equations
a+ (7/2) = 9
a=9-(7/2)
therefore a= 11/2
Given the a= 11/2 and b= 7/2,
a^4 + b^4
=(11/2)^4 + (7/2)^4
=(14641/16) + (2401/16)
=(8521/8)
a+b = 9 ...(1) .
a-b = 2....(2).
To find a^4+b^4.
(1)+(2): (a+b)+(a-b) = 2a = 9+2 = 11. So 2a/2 = 11/2 = 5.5. Or a = 5.5.
(1)-(2) : a+b -(a-b) = 2b = 9-2. So 2b = 7. Or b = 7/2 = 3.5.
Therefore a= 5.5 and b = 3.5.
=> a^4+b^4 = 5.5^4+3.5^4 = 1065.125.
Therefore a^4+b^4 = 5.5^.4+3.5^4 = 1065.125
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