Solve 2sqrt{16}+3sqrt[3]{16}+4sqrt[4]{16}+sqrt[6]{16}

$$2 \sqrt { 16 } + 3 \sqrt[ 3 ] { 16 } + 4 \sqrt[ 4 ] { 16 } + \sqrt[ 6 ] { 16 }$$

Evaluate

$$6\sqrt[3]{2}+2^{\frac{2}{3}}+16\approx 25.146927351$$

Solution Steps

Calculate the square root of 16 and get 4.

$$2\times 4+3\sqrt[3]{16}+4\sqrt[4]{16}+\sqrt[6]{16}$$

Multiply 2 and 4 to get 8.

$$8+3\sqrt[3]{16}+4\sqrt[4]{16}+\sqrt[6]{16}$$

Calculate \sqrt[4]{16} and get 2.

$$8+3\sqrt[3]{16}+4\times 2+\sqrt[6]{16}$$

Multiply 4 and 2 to get 8.

$$8+3\sqrt[3]{16}+8+\sqrt[6]{16}$$

Add 8 and 8 to get 16.

$$16+3\sqrt[3]{16}+\sqrt[6]{16}$$

Rewrite \sqrt[6]{16} as \sqrt[6]{4^{2}}. Convert from radical to exponential form and cancel out 2 in the exponent. Convert back to radical form.

$$\sqrt[6]{16}=\sqrt[6]{4^{2}}=4^{\frac{2}{6}}=4^{\frac{1}{3}}=\sqrt[3]{4}$$

Insert the obtained value back in the expression.

$$16+3\sqrt[3]{16}+\sqrt[3]{4}$$

Factor

$$6\sqrt[3]{2}+2^{\frac{2}{3}}+16$$