If \(a\) and \(b\) are positive integers (where \(b \neq 0\)) such that \(a^b = 1331\), then what is the value of \((a+b)^{a-b-8}\)?
1
To solve, write 1331 as a power of a prime using prime factorisation, read off the base and exponent, and then use the rule that any non-zero number raised to the power zero equals 1.
Step 1 — \(1331 = 11 \times 11 \times 11 = 11^3\), so \(a = 11\) and \(b = 3\).
Step 2 — \(a + b = 11 + 3 = 14\) and \(a - b - 8 = 11 - 3 - 8 = 0\).
Step 3 — \((a+b)^{a-b-8} = 14^0 = 1\).
The value 0 would need a zero base, while 13 and 23 would need an exponent of 1 with a matching base; neither happens here. (The trivial reading \(a = 1331, b = 1\) gives an enormous power that matches no choice.)
1 option is correct.
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