Discussion
If \( 2^{x} = 32 \), what is the value of \( x \)?
Answer: C
Correct answer: 5
Express 32 as a power of 2: \[ 32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^{5}. \] Since the bases are equal, the exponents must be equal, so \( x = 5 \). The rule being used is that \( a^{m} = a^{n} \) implies \( m = n \), provided \( a \) is positive and not 1. The powers of 2 are worth knowing by heart, since they appear constantly in both aptitude and computing questions: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. The laws of indices that go with this are the other things to have ready: \[ a^{m} \times a^{n} = a^{m+n}, \quad \frac{a^{m}}{a^{n}} = a^{m-n}, \quad (a^{m})^{n} = a^{mn}, \quad a^{0} = 1, \quad a^{-n} = \frac{1}{a^{n}}. \] The distractor 16 is 32 halved, offered for a reader who confuses the exponent with the value.
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