Answer:
1. [tex]81^{\frac{1}{4}}[/tex]
[tex]81^{\frac{1}{4}}=\sqrt[4]{81} =\boxed{\boldsymbol{3}}\\or\\81^{\frac{1}{4}}=(3^4)^{\frac{1}{4}}=3^{\frac{4}{4}}=\boxed{\boldsymbol{3}}[/tex]
2. [tex]121^{\frac{1}{2}}+256^{\frac{1}{4}}[/tex]
[tex]121^{\frac{1}{2}}+256^{\frac{1}{4}}=\sqrt{121}+\sqrt[4]{256}=11+4=\boxed{\boldsymbol{15}}\\or\\121^{\frac{1}{2}}+256^{\frac{1}{4}}=(11^2)^{\frac{1}{2}}+(4^4)^{\frac{1}{4}}=11^{\frac{2}{2}}+4^{\frac{4}{4}}=11+4=\boxed{\boldsymbol{15}}[/tex]
3. [tex]3,125^{\frac{2}{5}}[/tex]
[tex]3,125^{\frac{2}{5}}=(3,125^2)^{\frac{1}{5}}=\sqrt[5]{3,125^2}=\sqrt[5]{9,765,625}=\boxed{\boldsymbol{25}}\\or\\3,125^{\frac{2}{5}}=(5^5)^{\frac{2}{5}}=5^{\frac{10}{5}}=5^2=\boxed{\boldsymbol{25}}[/tex]
4. [tex]144^{\frac{1}{2}}-27^{\frac{1}{3}}[/tex]
[tex]144^{\frac{1}{2}}-27^{\frac{1}{3}}=\sqrt{144}-\sqrt[3]{27}=12-3=\boxed{\boldsymbol{9}}\\or\\144^{\frac{1}{2}}-27^{\frac{1}{3}}=(12^2)^{\frac{1}{2}}-(3^3)^{\frac{1}{3}}=12^{\frac{2}{2}}-3^{\frac{3}{3}}=12-3=\boxed{\boldsymbol{9}}[/tex]
5. [tex]32^{\frac{1}{5}}+9^{\frac{1}{2}}[/tex]
[tex]32^{\frac{1}{5}}+9^{\frac{1}{2}}=\sqrt[5]{32}+\sqrt{9}=2+3=\boxed{\boldsymbol{5}}\\or\\32^{\frac{1}{5}}+9^{\frac{1}{2}}=(2^5)^{\frac{1}{5}}+(3^2)^{\frac{1}{2}}=2^{\frac{5}{5}}+3^{\frac{2}{2}}=2+3=\boxed{\boldsymbol{5}}[/tex]
6. [tex](a^4b^{20})^{\frac{1}{4}}[/tex]
[tex](a^4b^{20})^{\frac{1}{4}}=\sqrt[4]{a^4b^{20}}=\boxed{\boldsymbol{ab^5}}\\or\\(a^4b^{20})^{\frac{1}{4}}=a^{\frac{4}{4}}b^{\frac{20}{4}}=\boxed{\boldsymbol{ab^5}}[/tex]
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