1. The problem is to evaluate fractional exponents and match them with their numerical values. 2. Recall the rule for fractional exponents: $$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$ where $a$ is the base, $m$ is the numerator (power), and $n$ is the denominator (root). 3. Evaluate each fractional exponent: - $4^{5/2} = (\sqrt{4})^5 = 2^5 = 32$ - $4^{3/2} = (\sqrt{4})^3 = 2^3 = 8$ - $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$ - $125^{1/3} = \sqrt[3]{125} = 5$ - $25^{1/2} = \sqrt{25} = 5$ - $(1/25)^{-1/2} = (25)^{1/2} = 5$ - $32^{2/5} = (\sqrt[5]{32})^2 = 2^2 = 4$ - $32^{6/5} = 32^{1 + 1/5} = 32 \times 32^{1/5} = 32 \times 2 = 64$ - $32^{4/5} = (\sqrt[5]{32})^4 = 2^4 = 16$ - $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$ - $9^{3/2} = (\sqrt{9})^3 = 3^3 = 27$ - $512^{2/3} = (\sqrt[3]{512})^2 = 8^2 = 64$ - $8^{4/3} = (\sqrt[3]{8})^4 = 2^4 = 16$ - $8^{5/3} = (\sqrt[3]{8})^5 = 2^5 = 32$ - $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$ - $64^{1/2} = \sqrt{64} = 8$ - $64^{2/3} = (\sqrt[3]{64})^2 = 4^2 = 16$ - $64^{5/6} = (\sqrt[6]{64})^5 = 2^5 = 32$ - $81^{3/4} = (\sqrt[4]{81})^3 = 3^3 = 27$ - $81^{2/4} = 81^{1/2} = \sqrt{81} = 9$ - $(1/81)^{-1/2} = 81^{1/2} = 9$ - $16^{3/2} = (\sqrt{16})^3 = 4^3 = 64$ - $16^{1/2} = \sqrt{16} = 4$ - $16^{1/4} = \sqrt[4]{16} = 2$ - $4^{1/2} = \sqrt{4} = 2$ - $8^{1/3} = \sqrt[3]{8} = 2$ - $243^{3/5} = (\sqrt[5]{243})^3 = 3^3 = 27$ - $81^{1/4} = \sqrt[4]{81} = 3$ - $27^{1/3} = \sqrt[3]{27} = 3$ - $(1/9)^{-1/2} = 9^{1/2} = 3$ 4. Match the evaluated values with the numbers given: - 2: $4^{1/2}$, $8^{1/3}$ - 3: $81^{1/4}$, $27^{1/3}$, $(1/9)^{-1/2}$ - 4: $32^{2/5}$, $8^{2/3}$, $16^{1/2}$ - 5: $125^{1/3}$, $25^{1/2}$, $(1/25)^{-1/2}$ - 8: $4^{3/2}$, $16^{3/4}$, $4^{5/2}$ (note $4^{5/2}=32$ so this is 32, corrected below) - 9: $27^{2/3}$, $9^{3/2}$, $(1/81)^{-1/2}$ - 10: $8^{4/3}$ (16) - 16: $32^{4/5}$, $64^{2/3}$, $8^{4/3}$ - 27: $81^{3/4}$, $243^{3/5}$ - 32: $4^{5/2}$, $64^{5/6}$, $8^{5/3}$ - 64: $16^{3/2}$, $32^{6/5}$, $512^{2/3}$ 5. Final matches: - 2: $4^{1/2}$, $8^{1/3}$, $16^{1/4}$ - 3: $81^{1/4}$, $27^{1/3}$, $(1/9)^{-1/2}$ - 4: $32^{2/5}$, $8^{2/3}$, $16^{1/2}$ - 5: $125^{1/3}$, $25^{1/2}$, $(1/25)^{-1/2}$ - 8: $4^{3/2}$, $16^{3/4}$, $8^{2/3}$ - 9: $27^{2/3}$, $9^{3/2}$, $(1/81)^{-1/2}$ - 16: $8^{4/3}$, $32^{4/5}$, $64^{2/3}$ - 27: $81^{3/4}$, $243^{3/5}$, $27^{1/3}$ - 32: $4^{5/2}$, $64^{5/6}$, $8^{5/3}$ - 64: $16^{3/2}$, $32^{6/5}$, $512^{2/3}$ q_count is 1 because this is one combined matching problem.