Solve 3/2sqrt{4^2*6}+sqrt[4]{2*3^4*3}-2sqrt[4]{2*2}+sqrt[4]{3^4*3}

$$\frac { 3 } { 2 } \sqrt { 4 ^ { 2 } \times 6 } + \sqrt[ 4 ] { 2 \cdot 3 ^ { 4 } \cdot 3 } - 2 \sqrt[ 4 ] { 2 \cdot 2 } + \sqrt[ 4 ] { 3 ^ { 4 } \cdot 3 }$$

Evaluate

$$3\sqrt[4]{6}+6\sqrt{6}+3^{\frac{5}{4}}-2\sqrt{2}\approx 20.511987111$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{3}{2}\sqrt{4^{2}\times 6}+\sqrt[4]{2\times 3^{5}}-2\sqrt[4]{2\times 2}+\sqrt[4]{3^{4}\times 3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{3}{2}\sqrt{4^{2}\times 6}+\sqrt[4]{2\times 3^{5}}-2\sqrt[4]{2\times 2}+\sqrt[4]{3^{5}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{3}{2}\sqrt{16\times 6}+\sqrt[4]{2\times 3^{5}}-2\sqrt[4]{2\times 2}+\sqrt[4]{3^{5}}$$

Multiply 16 and 6 to get 96.

$$\frac{3}{2}\sqrt{96}+\sqrt[4]{2\times 3^{5}}-2\sqrt[4]{2\times 2}+\sqrt[4]{3^{5}}$$

Factor 96=4^{2}\times 6. Rewrite the square root of the product \sqrt{4^{2}\times 6} as the product of square roots \sqrt{4^{2}}\sqrt{6}. Take the square root of 4^{2}.

$$\frac{3}{2}\times 4\sqrt{6}+\sqrt[4]{2\times 3^{5}}-2\sqrt[4]{2\times 2}+\sqrt[4]{3^{5}}$$

Multiply \frac{3}{2} and 4 to get 6.

$$6\sqrt{6}+\sqrt[4]{2\times 3^{5}}-2\sqrt[4]{2\times 2}+\sqrt[4]{3^{5}}$$

Calculate 3 to the power of 5 and get 243.

$$6\sqrt{6}+\sqrt[4]{2\times 243}-2\sqrt[4]{2\times 2}+\sqrt[4]{3^{5}}$$

Multiply 2 and 243 to get 486.

$$6\sqrt{6}+\sqrt[4]{486}-2\sqrt[4]{2\times 2}+\sqrt[4]{3^{5}}$$

Multiply 2 and 2 to get 4.

$$6\sqrt{6}+\sqrt[4]{486}-2\sqrt[4]{4}+\sqrt[4]{3^{5}}$$

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

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

Insert the obtained value back in the expression.

$$6\sqrt{6}+\sqrt[4]{486}-2\sqrt{2}+\sqrt[4]{3^{5}}$$

Calculate 3 to the power of 5 and get 243.

$$6\sqrt{6}+\sqrt[4]{486}-2\sqrt{2}+\sqrt[4]{243}$$

Factor

$$3 \sqrt[4]{3} + 3 \sqrt[4]{6} + 6 \sqrt{6} - 2 \sqrt{2} = 20.511987111$$