1. **Problem:** Simplify each algebraic expression involving multiplication and division of monomials with variables $m$ and $n$. 2. **Formula and rules:** - When multiplying monomials, multiply coefficients and add exponents of like bases: $a^x \cdot a^y = a^{x+y}$. - When dividing monomials, divide coefficients and subtract exponents of like bases: $\frac{a^x}{a^y} = a^{x-y}$. - Negative signs multiply normally. 3. **Step-by-step solutions:** **(a)** $(4mn)(-2mn) + \frac{24m^3n^2}{6m}$ - Multiply: $4 \times -2 = -8$ - Add exponents for $m$: $1+1=2$, for $n$: $1+1=2$ - So, $(4mn)(-2mn) = -8m^2n^2$ - Divide: $\frac{24m^3n^2}{6m} = 4m^{3-1}n^2 = 4m^2n^2$ - Sum: $-8m^2n^2 + 4m^2n^2 = -4m^2n^2$ **(b)** $\frac{-56m^7n^3}{8m^2n^2} - (5m^4n)(-4m)$ - Divide coefficients: $-56/8 = -7$ - Subtract exponents: $m^{7-2} = m^5$, $n^{3-2} = n^1$ - So, $\frac{-56m^7n^3}{8m^2n^2} = -7m^5n$ - Multiply: $5 \times -4 = -20$ - Multiply variables: $m^{4+1}n^{1} = m^5n$ - So, $(5m^4n)(-4m) = -20m^5n$ - Subtract: $-7m^5n - (-20m^5n) = -7m^5n + 20m^5n = 13m^5n$ **(c)** $\frac{(30m^3n^2)(10m^2n^5)}{(-2mn)(5mn)}$ - Numerator coefficients: $30 \times 10 = 300$ - Numerator variables: $m^{3+2}n^{2+5} = m^5n^7$ - Denominator coefficients: $-2 \times 5 = -10$ - Denominator variables: $m^{1+1}n^{1+1} = m^2n^2$ - Divide coefficients: $\frac{300}{-10} = -30$ - Divide variables: $m^{5-2}n^{7-2} = m^3n^5$ - Result: $-30m^3n^5$ **(d)** $\left(\frac{63m^6n^4}{-9mn^2}\right)(2m^2n)(-3m^3n)$ - Divide coefficients: $63 / -9 = -7$ - Divide variables: $m^{6-1}n^{4-2} = m^5n^2$ - Multiply by $2m^2n$: coefficients $-7 \times 2 = -14$, variables $m^{5+2}n^{2+1} = m^7n^3$ - Multiply by $-3m^3n$: coefficients $-14 \times -3 = 42$, variables $m^{7+3}n^{3+1} = m^{10}n^4$ - Result: $42m^{10}n^4$ **(e)** $\frac{(4mn)(-2mn^2)(-16m^5n^7)}{4m^4n^2}$ - Multiply numerator coefficients: $4 \times -2 \times -16 = 128$ - Multiply numerator variables: $m^{1+1+5}n^{1+2+7} = m^7n^{10}$ - Denominator: $4m^4n^2$ - Divide coefficients: $128 / 4 = 32$ - Divide variables: $m^{7-4}n^{10-2} = m^3n^8$ - Result: $32m^3n^8$ **(f)** $\frac{28m^5n^7}{-4mn^3} \times \frac{30mn^6}{6mn^4}$ - First fraction: coefficients $28 / -4 = -7$, variables $m^{5-1}n^{7-3} = m^4n^4$ - Second fraction: coefficients $30 / 6 = 5$, variables $m^{1-1}n^{6-4} = m^0n^2 = n^2$ - Multiply results: coefficients $-7 \times 5 = -35$, variables $m^4 \times m^0 = m^4$, $n^4 \times n^2 = n^6$ - Result: $-35m^4n^6$ **Final answers:** - (a) $-4m^2n^2$ - (b) $13m^5n$ - (c) $-30m^3n^5$ - (d) $42m^{10}n^4$ - (e) $32m^3n^8$ - (f) $-35m^4n^6$