1. Stating the problem: Simplify expressions involving roots and powers similar to the given examples. 2. Example 1: Simplify $$\sqrt[3]{8} \cdot \sqrt{16}$$. - Recall that $$\sqrt[3]{8} = 8^{\frac{1}{3}}$$ and $$\sqrt{16} = 16^{\frac{1}{2}}$$. - Calculate each: $$8^{\frac{1}{3}} = 2$$ since $$2^3=8$$, and $$16^{\frac{1}{2}}=4$$ since $$4^2=16$$. - Multiply: $$2 \cdot 4 = 8$$. 3. Example 2: Simplify $$2^4 \cdot 2^3 \cdot (-4)^2$$. - Use the rule $$a^m \cdot a^n = a^{m+n}$$ for same bases. - Combine powers of 2: $$2^{4+3} = 2^7 = 128$$. - Calculate $$(-4)^2 = 16$$. - Multiply: $$128 \cdot 16 = 2048$$. 4. Example 3: Simplify $$\frac{3^5 \cdot (-3)^3 \cdot 9}{3^4 \cdot 3^2}$$. - Express all terms with base 3: $$(-3)^3 = -27$$, $$9 = 3^2$$. - Numerator: $$3^5 \cdot (-27) \cdot 3^2 = 3^{5+2} \cdot (-27) = 3^7 \cdot (-27)$$. - Denominator: $$3^{4+2} = 3^6$$. - Divide powers: $$\frac{3^7}{3^6} = 3^{7-6} = 3^1 = 3$$. - Multiply by $$-27$$: $$3 \cdot (-27) = -81$$. 5. Example 4: Simplify $$ (5x^2 y^{-1})^3 \cdot (10x^{-3} y^4)^{-1} $$. - Apply power to each factor: $$5^3 x^{2 \cdot 3} y^{-1 \cdot 3} = 125 x^6 y^{-3}$$. - Inverse powers: $$10^{-1} x^{3} y^{-4}$$. - Multiply: $$125 \cdot 10^{-1} = 12.5$$. - Combine powers: $$x^{6+3} = x^9$$, $$y^{-3 + (-4)} = y^{-7}$$. - Final expression: $$12.5 x^9 y^{-7}$$. These examples illustrate simplifying roots, powers, and expressions with positive and negative exponents using exponent rules.