Incorrect exponent problems on a math worksheet; students must find errors, correct them, and explain their reasoning.

a) Mistake: Incorrectly calculated the exponent.
Correct: $2^5 = 32$
Justification: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$

b) Mistake: Incorrectly applied the exponent to the negative sign.
Correct: $(-2)^3 = -8$
Justification: $(-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8$

c) Mistake: Incorrectly applied the exponent to the negative sign.
Correct: $-6^2 = -36$
Justification: Exponentiation takes precedence over negation, so $-6^2 = -(6^2) = -36$

d) Mistake: Incorrectly assumed $x^0 = 0$ for all $x$.
Correct: $x^0 = 1$ for $x \neq 0$
Justification: Any non-zero number raised to the power of 0 is 1.

e) Mistake: Incorrectly multiplied the exponents.
Correct: $x^3 \cdot x^4 = x^{3+4} = x^7$
Justification: When multiplying like bases, add the exponents.

f) Mistake: Incorrectly subtracted the exponents.
Correct: $\frac{x^{10}}{x^5} = x^{10-5} = x^5$
Justification: When dividing like bases, subtract the exponents.

g) Mistake: Incorrectly multiplied the exponents.
Correct: $(x^3)^5 = x^{3 \cdot 5} = x^{15}$
Justification: When raising a power to a power, multiply the exponents.

h) Mistake: Incorrectly applied the negative exponent.
Correct: $7^{-2} = \frac{1}{7^2} = \frac{1}{49}$
Justification: A negative exponent means the reciprocal of the base raised to the positive exponent.

i) Mistake: Incorrectly applied the exponent to the negative sign.
Correct: $(-3)^4 = 81$
Justification: $(-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 9 \times 9 = 81$

j) Mistake: Incorrectly subtracted the exponents.
Correct: $\frac{x^5}{x^9} = x^{5-9} = x^{-4} = \frac{1}{x^4}$
Justification: When dividing like bases, subtract the exponents. The result is a negative exponent, which can be written as a reciprocal.