Solve sqrt[6]{2^3}-5sqrt{18}+15sqrt{30}-9sqrt{72}

$$\sqrt[ 6 ] { 2 ^ { 3 } } - 5 \sqrt { 18 } + 15 \sqrt { 30 } - 9 \sqrt { 72 }$$

Evaluate

$$15\sqrt{30}-68\sqrt{2}\approx -14.008138616$$

Solution Steps

Calculate 2 to the power of 3 and get 8.

$$\sqrt[6]{8}-5\sqrt{18}+15\sqrt{30}-9\sqrt{72}$$

Rewrite \sqrt[6]{8} as \sqrt[6]{2^{3}}. Convert from radical to exponential form and cancel out 3 in the exponent. Convert back to radical form.

$$\sqrt[6]{8}=\sqrt[6]{2^{3}}=2^{\frac{3}{6}}=2^{\frac{1}{2}}=\sqrt{2}$$

Insert the obtained value back in the expression.

$$\sqrt{2}-5\sqrt{18}+15\sqrt{30}-9\sqrt{72}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\sqrt{2}-5\times 3\sqrt{2}+15\sqrt{30}-9\sqrt{72}$$

Multiply -5 and 3 to get -15.

$$\sqrt{2}-15\sqrt{2}+15\sqrt{30}-9\sqrt{72}$$

Combine \sqrt{2} and -15\sqrt{2} to get -14\sqrt{2}.

$$-14\sqrt{2}+15\sqrt{30}-9\sqrt{72}$$

Factor 72=6^{2}\times 2. Rewrite the square root of the product \sqrt{6^{2}\times 2} as the product of square roots \sqrt{6^{2}}\sqrt{2}. Take the square root of 6^{2}.

$$-14\sqrt{2}+15\sqrt{30}-9\times 6\sqrt{2}$$

Multiply -9 and 6 to get -54.

$$-14\sqrt{2}+15\sqrt{30}-54\sqrt{2}$$

Combine -14\sqrt{2} and -54\sqrt{2} to get -68\sqrt{2}.

$$-68\sqrt{2}+15\sqrt{30}$$