1. Problem: Determine how many of the given exponentiation problems are incorrect. 2. We will check each expression by applying the exponent rules: - Multiplication with same base: $a^m \cdot a^n = a^{m+n}$ - Division with same base: $\frac{a^m}{a^n} = a^{m-n}$ - Power of a power: $(a^m)^n = a^{m \cdot n}$ - Any number to the zero power: $a^0 = 1$ 3. Check each: a) $5^2 \cdot 5^7 = 5^{2+7} = 5^9$, given $5^5$ is wrong. b) $2^{-2} \cdot 2^4 = 2^{-2+4} = 2^2$, correct. c) $g^6 : g^3 = g^{6-3} = g^3$, correct. d) $\frac{4^3}{4^5} = 4^{3-5} = 4^{-2}$, correct. e) $(5^3)^5 = 5^{3 \cdot 5} = 5^{15}$, correct. f) $(2^3)^2 = 2^{3 \cdot 2} = 2^6$, correct. g) $x \cdot x^4 \cdot x^{-3} = x^{1+4-3} = x^2$, given $x^7$ is wrong. h) $b \cdot 6^4 \cdot b^{-5} = 6^4 \cdot b^{1-5} = 6^4 \cdot b^{-4}$, given $6$ is wrong. i) $a^i : a^i = a^{i - i} = a^0 = 1$, given $a^0$ is correct. j) $c^{37} : c^5 \cdot (c^2)^2 = c^{37-5} \cdot c^{4} = c^{32+4} = c^{36}$, given $c^3$ is wrong. k) $y^9 \cdot y^{-12} \cdot y^5 = y^{9-12+5} = y^{2}$, given $y^3$ is wrong. l) $(x^5)^{-2} \cdot (x^3)^4 = x^{-10} \cdot x^{12} = x^{2}$, correct. m) $\frac{12^7}{12^3} = 12^{7-3} = 12^4$, correct. n) $(5^2)^{-1} = 5^{-2} = \frac{1}{5^2}$, correct. o) $(x^5)^0 = x^{5 \cdot 0} = x^0 = 1$, correct. p) $(2^4)^{-1} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16}$, given $\frac{1}{8}$ is wrong. 4. Count wrong answers: a), g), h), j), k), p) are wrong. Final answer: 6 tasks are incorrect.