1. **State the problem:** We are given three expressions involving exponents and asked to find values of $m$ and $n$, and to simplify an expression. 2. **First problem:** Simplify $x^5 \times x^7 = x^m$. - Rule: When multiplying powers with the same base, add the exponents: $x^a \times x^b = x^{a+b}$. - Calculation: $m = 5 + 7 = 12$. 3. **Second problem:** Simplify $\frac{y^8}{y^3} = y^n$. - Rule: When dividing powers with the same base, subtract the exponents: $\frac{y^a}{y^b} = y^{a-b}$. - Calculation: $n = 8 - 3 = 5$. 4. **Third problem:** Simplify fully $(5a^4 r^2)^3$. - Rule: When raising a product to a power, raise each factor to that power: $(abc)^n = a^n b^n c^n$. - Calculation: - $5^3 = 125$ - $a^{4 \times 3} = a^{12}$ - $r^{2 \times 3} = r^6$ - Final simplified form: $125 a^{12} r^6$. **Final answers:** - $m = 12$ - $n = 5$ - $(5a^4 r^2)^3 = 125 a^{12} r^6$