Question

Представьте выражение в виде степени с основание b.
а) $b sqrt{b sqrt[3]{b} }$
б) $sqrt[3]{b sqrt[3]{b sqrt{b} } }$
в) $sqrt[4]{ b^{2} sqrt[3]{b sqrt{b} } }$

Answer 1

$а) : : b sqrt{b sqrt[3]{b} } = b times {b}^{ frac{1}{2} } times ( {b}^{ frac{1}{3} } ) ^{ frac{1}{2} } = \ = {b}^{1 + frac{1}{2} + frac{1}{6} } = {b}^{ frac{6 + 3 + 1}{6} } = {b}^{ frac{10}{6} } = {b}^{1 frac{2}{3} }$

$б) : : sqrt[3]{b sqrt[3]{b sqrt{b} } } = \ = {b}^{ frac{1}{3} } times {b}^{ frac{1}{3} times frac{1}{3} } times {b}^{ frac{1}{2} times frac{1}{3} times frac{1}{3} } = \ = {b}^{ frac{1}{3} } times {b}^{ frac{1}{9} } times {b}^{ frac{1}{18} } = \ = {b}^{ frac{1}{3} + frac{1}{9} + frac{1}{18} } = {b}^{ frac{6 + 2 + 1}{18} } = {b}^{ frac{9}{18} } = {b}^{ frac{1}{2} }$

$в) : : sqrt[4]{ b^{2} sqrt[3]{b sqrt{b} } } = \ = {b}^{ frac{2}{4} } times {b}^{ frac{1}{3} times frac{1}{4} } times {b}^{ frac{1}{2} times frac{1}{3} times frac{1}{4} } = \ = {b}^{ frac{1}{2} } times {b}^{ frac{1}{12} } times {b}^{ frac{1}{24} } = \ = {b}^{ frac{1}{2} + frac{1}{12} + frac{1}{24} } = {b}^{ frac{12 + 2 + 1}{24} } = \ = {b}^{ frac{15}{24} } = {b}^{ frac{5}{8} }$