Solve 4sqrt{20}-sqrt{45}+sqrt[4]{25}+sqrt[5]{125}

$$4 \sqrt { 20 } - \sqrt { 45 } + \sqrt[ 4 ] { 25 } + \sqrt[ 5 ] { 125 }$$

Evaluate

$$\sqrt{5}\left(\sqrt[10]{5}+6\right)\approx 16.042935669$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$4\times 2\sqrt{5}-\sqrt{45}+\sqrt[4]{25}+\sqrt[5]{125}$$

Multiply 4 and 2 to get 8.

$$8\sqrt{5}-\sqrt{45}+\sqrt[4]{25}+\sqrt[5]{125}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$8\sqrt{5}-3\sqrt{5}+\sqrt[4]{25}+\sqrt[5]{125}$$

Combine 8\sqrt{5} and -3\sqrt{5} to get 5\sqrt{5}.

$$5\sqrt{5}+\sqrt[4]{25}+\sqrt[5]{125}$$

Rewrite \sqrt[4]{25} as \sqrt[4]{5^{2}}. Convert from radical to exponential form and cancel out 2 in the exponent. Convert back to radical form.

$$\sqrt[4]{25}=\sqrt[4]{5^{2}}=5^{\frac{2}{4}}=5^{\frac{1}{2}}=\sqrt{5}$$

Insert the obtained value back in the expression.

$$5\sqrt{5}+\sqrt{5}+\sqrt[5]{125}$$

Combine 5\sqrt{5} and \sqrt{5} to get 6\sqrt{5}.

$$6\sqrt{5}+\sqrt[5]{125}$$