Here are the solutions for the expressions: e) Simplify $2^n \times (2^n)^3 \times 2^{2n}$ Step 1: Simplify the term $(2^n)^3$ using the exponent rule $(a^m)^n = a^{mn}$. $$(2^n)^3 = 2^{n \times 3} = 2^{3n}$$ Step 2: Substitute this back into the expression. $$2^n \times 2^{3n} \times 2^{2n}$$ Step 3: Combine the terms using the exponent rule $a^m \times a^n \times a^p = a^{m+n+p}$. $$2^n \times 2^{3n} \times 2^{2n} = 2^{n + 3n + 2n}$$ $$2^{n + 3n + 2n} = 2^{6n}$$ The simplified expression is $\boxed{2^{6n}}$. f) Simplify $3^m \times 27^m \times 9^{-m}$ Step 1: Express all bases as powers of 3. $$27 = 3^3$$ $$9 = 3^2$$ Step 2: Substitute these into the expression. $$3^m \times (3^3)^m \times (3^2)^{-m}$$ Step 3: Simplify the terms with powers of powers using $(a^x)^y = a^{xy}$. $$(3^3)^m = 3^{3m}$$ $$(3^2)^{-m} = 3^{2 \times (-m)} = 3^{-2m}$$ Step 4: Substitute these back into the expression. $$3^m \times 3^{3m} \times 3^{-2m}$$ Step 5: Combine the terms using the exponent rule $a^x \times a^y \times a^z = a^{x+y+z}$. $$3^m \times 3^{3m} \times 3^{-2m} = 3^{m + 3m + (-2m)}$$ $$3^{m + 3m - 2m} = 3^{4m - 2m}$$ $$3^{4m - 2m} = 3^{2m}$$ The simplified expression is $\boxed{3^{2m}}$. g) Simplify $(a^{1/2} \times a)^{2.5}$ Step 1: Simplify inside the parenthesis. Remember that $a = a^1$. Use the exponent rule $a^x \times a^y = a^{x+y}$. $$a^{1/2} \times a^1 = a^{1/2 + 1}$$ $$a^{1/2 + 1} = a^{1/2 + 2/2} = a^{3/2}$$ Step 2: Substitute this back into the expression. $$(a^{3/2})^{2.5}$$ Step 3: Convert the decimal exponent $2.5$ to a fraction. $$2.5 = \frac{25}{10} = \frac{5}{2}$$ Step 4: Apply the outer exponent using the rule $(a^x)^y = a^{xy}$. $$(a^{3/2})^{5/2} = a^{\frac{3}{2} \times \frac{5}{2}}$$ $$a^{\frac{3}{2} \times \frac{5}{2}} = a^{\frac{3 \times 5}{2 \times 2}}$$ $$a^{\frac{15}{4}}$$ The simplified expression is $\boxed{a^{\frac{15}{4}}}$.