If $5\sqrt{5} \times 5^3 \div 5^{-\frac{3}{2}} = 5^{a+2}$, then the value of $a$ is

### Concept & Formula This problem tests the application of multiple laws of indices simultaneously. Convert all radicals into fractional exponents, and combine terms with the same base using addition for multiplication and subtraction for division. Key rules applied: $$x^m \times x^n = x^{m+n}$$ $$x^m \div x^n = x^{m-n}$$ $$\sqrt{x} = x^{\frac{1}{2}}$$ ### Step-by-Step Solution * **Given:** $$5\sqrt{5} \times 5^3 \div 5^{-\frac{3}{2}} = 5^{a+2}$$ * **Calculation:** Convert the surd into an exponent: $5\sqrt{5} = 5^1 \times 5^{\frac{1}{2}} = 5^{1 + 0.5} = 5^{1.5}$ Rewrite the entire equation with explicit exponents: $$5^{1.5} \times 5^3 \div 5^{-1.5} = 5^{a+2}$$ Apply the product rule (add exponents for multiplication): $$5^{1.5 + 3} \div 5^{-1.5} = 5^{a+2}$$ $$5^{4.5} \div 5^{-1.5} = 5^{a+2}$$ Apply the quotient rule (subtract exponents for division): $$5^{4.5 - (-1.5)} = 5^{a+2}$$ $$5^{4.5 + 1.5} = 5^{a+2}$$ $$5^6 = 5^{a+2}$$ Equate the exponents since the bases are identical: $$6 = a + 2$$ $$a = 4$$ ### Exam Strategy & Shortcut **Work Only with Exponents:** Once you verify the base is uniformly $5$, completely drop the base and write a linear equation of only the exponents to save time: $$1 + 0.5 + 3 - (-1.5) = a + 2$$ $$4.5 + 1.5 = a + 2$$ $$6 = a + 2 \Rightarrow a = 4$$ This takes less than 10 seconds. ### Common Pitfall The most common error is mishandling the division of a negative exponent. Students often write $4.5 - 1.5 = 3$ instead of correctly subtracting the negative value $4.5 - (-1.5) = 6$. Always pay strict attention to double negatives. ### Final Answer **Therefore, the correct answer is 4.**