1. **State the problem:** Find the value of $n$ given the equation $$3^{\frac{3}{2}} = 3^7$$ and then solve $$3^{\frac{3}{4}} \times 3^7 = 3^{\frac{24}{4}}.$$ 2. **Recall the exponent rules:** When multiplying powers with the same base, add the exponents: $$a^m \times a^n = a^{m+n}.$$ 3. **Solve the first equation:** Since the bases are the same, set the exponents equal: $$\frac{3}{2} = 7.$$ This is not true, so likely the problem wants to find $n$ such that $$3^n = 3^7$$ or to check the consistency. 4. **Solve the second equation:** Use the rule for multiplication of powers: $$3^{\frac{3}{4}} \times 3^7 = 3^{\frac{3}{4} + 7} = 3^{\frac{3}{4} + \frac{28}{4}} = 3^{\frac{31}{4}}.$$ 5. **Compare with the right side:** The problem states this equals $$3^{\frac{24}{4}} = 3^6.$$ 6. **Set exponents equal:** $$\frac{31}{4} = 6.$$ This is false, so there might be a misunderstanding. 7. **Assuming the problem is to find $n$ such that:** $$3^{\frac{3}{4}} \times 3^n = 3^{\frac{24}{4}}.$$ Then add exponents: $$\frac{3}{4} + n = 6.$$ 8. **Solve for $n$:** $$n = 6 - \frac{3}{4} = \frac{24}{4} - \frac{3}{4} = \frac{21}{4} = 5.25.$$ --- **Part (b):** Find 6% of $$8.2 \times 10^{145}$$ and give the answer in standard form. 1. **Calculate 6% as a decimal:** $$6\% = 0.06.$$ 2. **Multiply:** $$0.06 \times 8.2 \times 10^{145} = 0.492 \times 10^{145}.$$ 3. **Convert to standard form:** $$0.492 \times 10^{145} = 4.92 \times 10^{-1} \times 10^{145} = 4.92 \times 10^{144}.$$ --- **Final answers:** - Value of $n$ is $$\boxed{\frac{21}{4}}$$ or 5.25. - 6% of $$8.2 \times 10^{145}$$ is $$\boxed{4.92 \times 10^{144}}.$$