Solve sqrt{75}-sqrt[9]{27}

$$\sqrt { 75 } - \sqrt[ 9 ] { 27 }$$

Evaluate

$$5\sqrt{3}-\sqrt[3]{3}\approx 7.218004468$$

Solution Steps

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$5\sqrt{3}-\sqrt[9]{27}$$

Rewrite \sqrt[9]{27} as \sqrt[9]{3^{3}}. Convert from radical to exponential form and cancel out 3 in the exponent. Convert back to radical form.

$$\sqrt[9]{27}=\sqrt[9]{3^{3}}=3^{\frac{3}{9}}=3^{\frac{1}{3}}=\sqrt[3]{3}$$

Insert the obtained value back in the expression.

$$5\sqrt{3}-\sqrt[3]{3}$$