In this question, x^y stands for x raised to the power y. For example, 2^3=8…
2025
In this question, x^y stands for x raised to the power y. For example, 2^3=8 and 4^1.5=8. If a,b are real numbers such that a+b=3, a^2+ b^2=7, the value of a^4+b^4 is?
- A.
47
- B.
51
- C.
45
- D.
49
Attempted by 5 students.
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Correct answer: A
Two identities link a+b, a2+b2, ab and a4+b4: squaring (a+b) gives (a+b)2 = a2+b2+2ab, so ab can be recovered once a+b and a2+b2 are known; and squaring (a2+b2) gives a4+b4 = (a2+b2)2 − 2(ab)2, so a4+b4 follows directly from a2+b2 and ab without solving for a and b individually.
Given a+b=3 and a2+b2=7, square the sum: (a+b)2 = a2+b2+2ab, so 32 = 7 + 2ab.
Simplify: 9 = 7 + 2ab, so 2ab = 2, giving ab = 1.
Apply the fourth-power identity: a4+b4 = (a2+b2)2 − 2(ab)2 = 72 − 2(1)2.
Compute the two terms: 72 = 49 and 2(1)2 = 2, so a4+b4 = 49 − 2 = 47.
Independent check: with a+b=3 and ab=1, a and b are the roots of t2 − 3t + 1 = 0, i.e. t = (3±√5)/2. Substituting back, a2+b2 = (a+b)2 − 2ab = 9 − 2 = 7, matching the given data, and evaluating a4+b4 numerically for these roots also gives 47 — confirming the identity-based result.