Discussion

If \(a \oplus y = a^{2} + 2y\), what is the value of p if \(8 \oplus (5 \oplus p) = 138\) ?

Answer: D

The value of p is 6.

Apply the rule \(a \oplus y = a^{2} + 2y\) from the inside out:

\[ \begin{aligned} 5 \oplus p &= 25 + 2p \\ 8 \oplus (25 + 2p) &= 64 + 2(25 + 2p) \\ &= 114 + 4p \end{aligned} \]

Setting this equal to 138:

\[ 114 + 4p = 138 \Rightarrow 4p = 24 \Rightarrow p = 6 \]

Check: \(5 \oplus 6 = 37\) and \(8 \oplus 37 = 64 + 74 = 138\). The other values give 134 (p = 5), 130 (p = 4) and 146 (p = 8).

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