Simplify \(\sqrt{\frac{7}{8}}\) to \(\frac{\sqrt{7}}{\sqrt{8}}\).
\[\sqrt{\frac{1}{2}\sqrt{\frac{5}{7}\times \frac{\sqrt{7}}{\sqrt{8}}}}\]
Simplify \(\sqrt{8}\) to \(2\sqrt{2}\).
\[\sqrt{\frac{1}{2}\sqrt{\frac{5}{7}\times \frac{\sqrt{7}}{2\sqrt{2}}}}\]
Rationalize the denominator: \(\frac{5}{7}\times \frac{\sqrt{7}}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{5\sqrt{7}\sqrt{2}}{7\times 2\times 2}\).
\[\sqrt{\frac{1}{2}\sqrt{\frac{5\sqrt{7}\sqrt{2}}{7\times 2\times 2}}}\]
Simplify \(5\sqrt{7}\sqrt{2}\) to \(5\sqrt{14}\).
\[\sqrt{\frac{1}{2}\sqrt{\frac{5\sqrt{14}}{7\times 2\times 2}}}\]
Simplify \(7\times 2\) to \(14\).
\[\sqrt{\frac{1}{2}\sqrt{\frac{5\sqrt{14}}{14\times 2}}}\]
Simplify \(14\times 2\) to \(28\).
\[\sqrt{\frac{1}{2}\sqrt{\frac{5\sqrt{14}}{28}}}\]
Simplify \(\sqrt{\frac{5\sqrt{14}}{28}}\) to \(\frac{\sqrt{5\sqrt{14}}}{\sqrt{28}}\).
\[\sqrt{\frac{1}{2}\times \frac{\sqrt{5\sqrt{14}}}{\sqrt{28}}}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\sqrt{\frac{1}{2}\times \frac{\sqrt{5}\sqrt{\sqrt{14}}}{\sqrt{28}}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\sqrt{\frac{1}{2}\times \frac{\sqrt{5}\times {14}^{\frac{1\times 1}{2\times 2}}}{\sqrt{28}}}\]
Simplify \(1\times 1\) to \(1\).
\[\sqrt{\frac{1}{2}\times \frac{\sqrt{5}\sqrt[2\times 2]{14}}{\sqrt{28}}}\]
Simplify \(2\times 2\) to \(4\).
\[\sqrt{\frac{1}{2}\times \frac{\sqrt{5}\sqrt[4]{14}}{\sqrt{28}}}\]
Regroup terms.
\[\sqrt{\frac{1}{2}\times \frac{\sqrt[4]{14}\sqrt{5}}{\sqrt{28}}}\]
Simplify \(\sqrt{28}\) to \(2\sqrt{7}\).
\[\sqrt{\frac{1}{2}\times \frac{\sqrt[4]{14}\sqrt{5}}{2\sqrt{7}}}\]
Rationalize the denominator: \(\frac{1}{2}\times \frac{\sqrt[4]{14}\sqrt{5}}{2\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt[4]{14}\sqrt{5}\sqrt{7}}{2\times 2\times 7}\).
\[\sqrt{\frac{\sqrt[4]{14}\sqrt{5}\sqrt{7}}{2\times 2\times 7}}\]
Simplify \(\sqrt[4]{14}\sqrt{5}\sqrt{7}\) to \(\sqrt[4]{14}\sqrt{35}\).
\[\sqrt{\frac{\sqrt[4]{14}\sqrt{35}}{2\times 2\times 7}}\]
Simplify \(2\times 2\) to \(4\).
\[\sqrt{\frac{\sqrt[4]{14}\sqrt{35}}{4\times 7}}\]
Simplify \(4\times 7\) to \(28\).
\[\sqrt{\frac{\sqrt[4]{14}\sqrt{35}}{28}}\]
Simplify.
\[\frac{\sqrt{\sqrt[4]{14}\sqrt{35}}}{\sqrt{28}}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{\sqrt{\sqrt[4]{14}}\sqrt{\sqrt{35}}}{\sqrt{28}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{14}^{\frac{1\times 1}{4\times 2}}\sqrt{\sqrt{35}}}{\sqrt{28}}\]
Simplify \(1\times 1\) to \(1\).
\[\frac{\sqrt[4\times 2]{14}\sqrt{\sqrt{35}}}{\sqrt{28}}\]
Simplify \(4\times 2\) to \(8\).
\[\frac{\sqrt[8]{14}\sqrt{\sqrt{35}}}{\sqrt{28}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{\sqrt[8]{14}\times {35}^{\frac{1\times 1}{2\times 2}}}{\sqrt{28}}\]
Simplify \(1\times 1\) to \(1\).
\[\frac{\sqrt[8]{14}\sqrt[2\times 2]{35}}{\sqrt{28}}\]
Simplify \(2\times 2\) to \(4\).
\[\frac{\sqrt[8]{14}\sqrt[4]{35}}{\sqrt{28}}\]
Simplify \(\sqrt{28}\) to \(2\sqrt{7}\).
\[\frac{\sqrt[8]{14}\sqrt[4]{35}}{2\sqrt{7}}\]
Rationalize the denominator: \(\frac{\sqrt[8]{14}\sqrt[4]{35}}{2\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt[8]{14}\sqrt[4]{35}\sqrt{7}}{2\times 7}\).
\[\frac{\sqrt[8]{14}\sqrt[4]{35}\sqrt{7}}{2\times 7}\]
Simplify \(2\times 7\) to \(14\).
\[\frac{\sqrt[8]{14}\sqrt[4]{35}\sqrt{7}}{14}\]
Decimal Form: 0.639299
(14^(1/8)*35^(1/4)*sqrt(7))/14
