1. **State the problem:** Simplify each algebraic expression involving exponents. 2. **Recall exponent rules:** - Product rule: $a^m \cdot a^n = a^{m+n}$ - Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$ - Power of a power: $(a^m)^n = a^{mn}$ - Negative exponent: $a^{-n} = \frac{1}{a^n}$ 3. **Simplify each expression:** - Expression 1: $4x$ This is already simplified. - Expression 2: $3a^2 b^5$ Already simplified. - Expression 3: $\frac{m^6 n}{w^7}$ Already simplified. - Expression 4: $x^4 y \cdot x^3 y^9$ Apply product rule: $$x^{4+3} y^{1+9} = x^7 y^{10}$$ - Expression 5: $-7y^2 \cdot 2y^{-5}$ Multiply coefficients: $-7 \times 2 = -14$ Apply product rule to $y$ terms: $$y^{2 + (-5)} = y^{-3}$$ So expression is: $$-14 y^{-3} = -\frac{14}{y^3}$$ - Expression 6: $\frac{10x^9}{5x^5}$ Simplify coefficients: $$\frac{10}{5} = 2$$ Apply quotient rule to $x$ terms: $$x^{9-5} = x^4$$ So expression is: $$2x^4$$ - Expression 7: $\frac{20x^7 y^5}{5x^2 y^5}$ Simplify coefficients: $$\frac{20}{5} = 4$$ Apply quotient rule to $x$ terms: $$x^{7-2} = x^5$$ Apply quotient rule to $y$ terms: $$y^{5-5} = y^0 = 1$$ So expression is: $$4x^5$$ **Final answers:** 1. $4x$ 2. $3a^2 b^5$ 3. $\frac{m^6 n}{w^7}$ 4. $x^7 y^{10}$ 5. $-\frac{14}{y^3}$ 6. $2x^4$ 7. $4x^5$