Numerical answer
Find the sum of all real values of x satisfying (x² − 7x + 11)^(x² − 13x + 42) = 1.
Show answer and solution
Answer: 27
aᵇ = 1 in three ways:
1. Exponent 0 (base ≠ 0): x² − 13x + 42 = 0 ⇒ x = 6, 7. Bases: 36 − 42 + 11 = 5 and 49 − 49 + 11 = 11, both non-zero ✓.
2. Base 1: x² − 7x + 11 = 1 ⇒ x² − 7x + 10 = 0 ⇒ x = 2, 5 ✓.
3. Base −1 with even exponent: x² − 7x + 12 = 0 ⇒ x = 3, 4. Exponents: 9 − 39 + 42 = 12 and 16 − 52 + 42 = 6, both even ✓.
The values are 2, 3, 4, 5, 6, 7 (all distinct); sum = 27.
Trap: students who only solve 'exponent = 0' and 'base = 1' get 20; the base −1 case is the one CAT likes to test.
1. Exponent 0 (base ≠ 0): x² − 13x + 42 = 0 ⇒ x = 6, 7. Bases: 36 − 42 + 11 = 5 and 49 − 49 + 11 = 11, both non-zero ✓.
2. Base 1: x² − 7x + 11 = 1 ⇒ x² − 7x + 10 = 0 ⇒ x = 2, 5 ✓.
3. Base −1 with even exponent: x² − 7x + 12 = 0 ⇒ x = 3, 4. Exponents: 9 − 39 + 42 = 12 and 16 − 52 + 42 = 6, both even ✓.
The values are 2, 3, 4, 5, 6, 7 (all distinct); sum = 27.
Trap: students who only solve 'exponent = 0' and 'base = 1' get 20; the base −1 case is the one CAT likes to test.
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