1. Evaluate the expression $(-7)^2 \cdot (-7)^2$. Step 1: Recall the rule for powers: $a^m \cdot a^n = a^{m+n}$. Step 2: Apply the rule: $$(-7)^2 \cdot (-7)^2 = (-7)^{2+2} = (-7)^4$$ Step 3: Calculate $(-7)^4$: $$(-7)^4 = ((-7)^2)^2 = (49)^2 = 2401$$ Final answer: $2401$. 2. Evaluate $6^{-2} \cdot \frac{1}{36}$. Step 1: Recall $a^{-n} = \frac{1}{a^n}$. Step 2: Rewrite $6^{-2}$: $$6^{-2} = \frac{1}{6^2} = \frac{1}{36}$$ Step 3: Multiply: $$\frac{1}{36} \cdot \frac{1}{36} = \frac{1}{36^2} = \frac{1}{1296}$$ Final answer: $\frac{1}{1296}$. 3. Evaluate $8^0 \cdot 2^4$. Step 1: Recall $a^0 = 1$ for any $a \neq 0$. Step 2: Calculate: $$8^0 = 1$$ $$2^4 = 16$$ Step 3: Multiply: $$1 \cdot 16 = 16$$ Final answer: $16$. 4. Simplify $\frac{3^2 \cdot 3^{-1}}{3^5}$ using only positive exponents. Step 1: Use the product rule: $$3^2 \cdot 3^{-1} = 3^{2 + (-1)} = 3^1 = 3$$ Step 2: Write the expression: $$\frac{3}{3^5} = 3^{1 - 5} = 3^{-4}$$ Step 3: Rewrite with positive exponent: $$3^{-4} = \frac{1}{3^4} = \frac{1}{81}$$ Final answer: $\frac{1}{81}$. 5. Simplify $\frac{4^2 x^{-7}}{x^{-3}}$ using only positive exponents. Step 1: Simplify the $x$ terms using quotient rule: $$\frac{x^{-7}}{x^{-3}} = x^{-7 - (-3)} = x^{-7 + 3} = x^{-4}$$ Step 2: Write the expression: $$4^2 \cdot x^{-4} = 16 \cdot x^{-4}$$ Step 3: Rewrite with positive exponent: $$16 \cdot \frac{1}{x^4} = \frac{16}{x^4}$$ Final answer: $\frac{16}{x^4}$. 6. Simplify $(-4x^{-2}) \cdot (-2x^{-3} y^3)$ using only positive exponents. Step 1: Multiply coefficients: $$-4 \cdot -2 = 8$$ Step 2: Multiply $x$ terms: $$x^{-2} \cdot x^{-3} = x^{-2 + (-3)} = x^{-5}$$ Step 3: Multiply $y$ terms: $$y^3$$ Step 4: Combine: $$8 \cdot x^{-5} \cdot y^3 = 8 y^3 x^{-5}$$ Step 5: Rewrite with positive exponent: $$8 y^3 \cdot \frac{1}{x^5} = \frac{8 y^3}{x^5}$$ Final answer: $\frac{8 y^3}{x^5}$. 7. Simplify $\frac{3 n^{-2}}{n^3}$ using only positive exponents. Step 1: Use quotient rule for $n$: $$\frac{n^{-2}}{n^3} = n^{-2 - 3} = n^{-5}$$ Step 2: Write expression: $$3 \cdot n^{-5} = 3 n^{-5}$$ Step 3: Rewrite with positive exponent: $$3 \cdot \frac{1}{n^5} = \frac{3}{n^5}$$ Final answer: $\frac{3}{n^5}$. 8. Simplify $\frac{z^3 \cdot z^{-5}}{4}$ using only positive exponents. Step 1: Multiply $z$ terms: $$z^{3 + (-5)} = z^{-2}$$ Step 2: Write expression: $$\frac{z^{-2}}{4} = \frac{1}{4} \cdot z^{-2}$$ Step 3: Rewrite with positive exponent: $$\frac{1}{4} \cdot \frac{1}{z^2} = \frac{1}{4 z^2}$$ Final answer: $\frac{1}{4 z^2}$.