(1) \( \frac{25^{-5} \cdot 15^{x+1}}{3^{x} \cdot 5^{-5}} \) 2) \( (3 m-2 n)(m+2 n-5) \)

Simplify the expression by following steps: - step0: Multiply the terms: \(\left(3m-2n\right)\left(m+2n-5\right)\) - step1: Apply the distributive property: \(3m\times m+3m\times 2n-3m\times 5-2nm-2n\times 2n-\left(-2n\times 5\right)\) - step2: Multiply the terms: \(3m^{2}+6mn-15m-2nm-4n^{2}-\left(-10n\right)\) - step3: Remove the parentheses: \(3m^{2}+6mn-15m-2nm-4n^{2}+10n\) - step4: Subtract the terms: \(3m^{2}+4mn-15m-4n^{2}+10n\) Calculate or simplify the expression \( \frac{25^{-5} \cdot 15^{x+1}}{3^{x} \cdot 5^{-5}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{25^{-5}\times 15^{x+1}}{3^{x}\times 5^{-5}}\) - step1: Factor the expression: \(\frac{5^{-10}\times 15^{x+1}}{3^{x}\times 5^{-5}}\) - step2: Reduce the fraction: \(\frac{15^{x+1}}{3^{x}\times 5^{5}}\) - step3: Factor the expression: \(\frac{3^{x+1}\times 5^{x+1}}{3^{x}\times 5^{5}}\) - step4: Reduce the fraction: \(3\times 5^{x-4}\) Let's break down the solutions for both expressions step by step. ### 1) Simplifying \( \frac{25^{-5} \cdot 15^{x+1}}{3^{x} \cdot 5^{-5}} \) **Step 1: Rewrite the bases.** - \( 25 = 5^2 \), so \( 25^{-5} = (5^2)^{-5} = 5^{-10} \). - \( 15 = 3 \cdot 5 \), so \( 15^{x+1} = (3 \cdot 5)^{x+1} = 3^{x+1} \cdot 5^{x+1} \). **Step 2: Substitute back into the expression.** \[ \frac{5^{-10} \cdot (3^{x+1} \cdot 5^{x+1})}{3^{x} \cdot 5^{-5}} = \frac{3^{x+1} \cdot 5^{-10 + x + 1}}{3^{x} \cdot 5^{-5}} \] **Step 3: Simplify the powers of 5.** \[ = \frac{3^{x+1}}{3^{x}} \cdot 5^{-10 + x + 1 + 5} = 3^{(x+1) - x} \cdot 5^{-4 + x} = 3^{1} \cdot 5^{x - 4} \] **Final Result:** \[ 3 \cdot 5^{x - 4} \] ### 2) Expanding \( (3 m-2 n)(m+2 n-5) \) **Step 1: Use the distributive property (FOIL method).** - First: \( 3m \cdot m = 3m^2 \) - Outer: \( 3m \cdot 2n = 6mn \) - Inner: \( -2n \cdot m = -2mn \) - Last: \( -2n \cdot -5 = 10n \) **Step 2: Combine like terms.** \[ 3m^2 + 6mn - 2mn + 10n = 3m^2 + (6mn - 2mn) + 10n = 3m^2 + 4mn + 10n \] **Final Result:** \[ 3m^2 + 4mn - 15m - 4n^2 + 10n \] ### Summary of Results: 1) \( \frac{25^{-5} \cdot 15^{x+1}}{3^{x} \cdot 5^{-5}} = 3 \cdot 5^{x - 4} \) 2) \( (3 m-2 n)(m+2 n-5) = 3m^2 + 4mn - 15m - 4n^2 + 10n \)