1. Simplify $w^5 w^{-8} w^4$. Step 1: State the problem: Simplify the product of powers with the same base $w$. Step 2: Use the product of powers rule: $$a^m \cdot a^n = a^{m+n}$$ Step 3: Add the exponents: $$5 + (-8) + 4 = 5 - 8 + 4 = 1$$ Step 4: So, $$w^5 w^{-8} w^4 = w^{1} = w$$ Final answer: $w$ 2. Simplify $\left(\frac{y^2}{x}\right)^3$. Step 1: State the problem: Simplify the power of a fraction. Step 2: Use the power of a quotient rule: $$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$$ Step 3: Apply the rule: $$\left(\frac{y^2}{x}\right)^3 = \frac{(y^2)^3}{x^3}$$ Step 4: Simplify numerator: $$(y^2)^3 = y^{2 \cdot 3} = y^6$$ Step 5: Final expression: $$\frac{y^6}{x^3}$$ 3. Simplify $a^0$. Step 1: State the problem: Simplify any nonzero base raised to zero. Step 2: Use the zero exponent rule: $$a^0 = 1, \text{ for } a \neq 0$$ Step 3: Final answer: $1$ 4. Simplify $x^2 y^3 z^4 (x^3 y^0 z^2)^2$. Step 1: State the problem: Simplify the product of powers and powers of powers. Step 2: Use power of a power rule: $$(a^m)^n = a^{m \cdot n}$$ Step 3: Simplify inside the parentheses: $y^0 = 1$ Step 4: Apply power of power: $$(x^3 y^0 z^2)^2 = x^{3 \cdot 2} y^{0 \cdot 2} z^{2 \cdot 2} = x^6 y^0 z^4 = x^6 z^4$$ Step 5: Multiply the original terms: $$x^2 y^3 z^4 \cdot x^6 z^4 = x^{2+6} y^3 z^{4+4} = x^8 y^3 z^8$$ 5. Simplify $(x^5)^6$. Step 1: Use power of a power rule: $$(x^5)^6 = x^{5 \cdot 6} = x^{30}$$ 6. Simplify $3x^2 \cdot (4x^3)^2$. Step 1: Simplify inside the parentheses: $$(4x^3)^2 = 4^2 (x^3)^2 = 16 x^{6}$$ Step 2: Multiply: $$3x^2 \cdot 16 x^6 = 48 x^{2+6} = 48 x^8$$ 7. Simplify $6^7 \cdot 6^9$. Step 1: Use product of powers rule: $$6^{7+9} = 6^{16}$$ 8. Simplify $(-2x)^4$. Step 1: Apply power to product: $$(-2)^4 x^4 = 16 x^4$$ Final answers: 1. $w$ 2. $\frac{y^6}{x^3}$ 3. $1$ 4. $x^8 y^3 z^8$ 5. $x^{30}$ 6. $48 x^8$ 7. $6^{16}$ 8. $16 x^4$