Advanced Math practice question

Answer: 14/3 or 4.66 or 4.67

Write every factor as a power of 2: (1/8)^x = (2^(−3))^x = 2^(−3x); 4^(2x + 1) = (2²)^(2x + 1) = 2^(4x + 2); 16^(x − 3) = (2⁴)^(x − 3) = 2^(4x − 12). The left side becomes 2^(−3x) · 2^(4x + 2) = 2^(x + 2). Since the bases are equal and 2^t is one-to-one, x + 2 = 4x − 12, so 3x = 14 and x = 14/3 (≈ 4.67). Common errors this item targets: writing (1/8)^x as 2^(3x) instead of 2^(−3x) (sign error on the reciprocal), which yields 7x + 2 = 4x − 12 and x = −14/3; multiplying rather than adding exponents on the left, giving −3x(4x + 2) and a spurious quadratic; and distributing the outer exponent incorrectly as 16^(x − 3) = 2^(4x − 3), which gives x + 2 = 4x − 3 and x = 5/3.

Common wrong answers

If you answered 14

You stopped at 3x = 14 — divide by 3 to get x = 14/3.

If you answered -14/3

You wrote (1/8)^x as 2^(3x) instead of 2^(−3x); the reciprocal makes the exponent negative, so the left side is 2^(x + 2), not 2^(7x + 2).

If you answered 5/3

You distributed the outer exponent incorrectly, treating 16^(x − 3) as 2^(4x − 3); multiply the whole exponent by 4 to get 2^(4x − 12).

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