1. **Problem:** Simplify $x^{12} \times x^3$. 2. **Rule:** When multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$. 3. **Work:** $$x^{12} \times x^3 = x^{12+3} = x^{15}$$ 4. **Answer:** $x^{15}$ (Option A). 1. **Problem:** Simplify $\frac{x^3}{x^{12}}$. 2. **Rule:** When dividing powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. 3. **Work:** $$\frac{x^3}{x^{12}} = x^{3-12} = x^{-9} = \frac{1}{x^9}$$ 4. **Answer:** $\frac{1}{x^9}$ (Option B). 1. **Problem:** Simplify $(x^3)^{12}$. 2. **Rule:** When raising a power to another power, multiply the exponents: $(a^m)^n = a^{m \times n}$. 3. **Work:** $$(x^3)^{12} = x^{3 \times 12} = x^{36}$$ 4. **Answer:** $x^{36}$ (Option B). 1. **Problem:** Simplify $18x^0$. 2. **Rule:** Any nonzero number raised to the zero power is 1: $a^0 = 1$. 3. **Work:** $$18x^0 = 18 \times 1 = 18$$ 4. **Answer:** $18$ (none of the given options A or B or C or D match exactly, but the simplification is $18$).