Solve sqrt[9]{27}+sqrt[15]{243}-sqrt[6]{9}

$$\sqrt[ 9 ]{ 27 } + \sqrt[ 15 ]{ 243 } - \sqrt[ 6 ]{ 9 }$$

Evaluate

$$\sqrt[3]{3}\approx 1.44224957$$

Solution Steps

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.

$$\sqrt[3]{3}+\sqrt[15]{243}-\sqrt[6]{9}$$

Rewrite \sqrt[15]{243} as \sqrt[15]{3^{5}}. Convert from radical to exponential form and cancel out 5 in the exponent. Convert back to radical form.

$$\sqrt[15]{243}=\sqrt[15]{3^{5}}=3^{\frac{5}{15}}=3^{\frac{1}{3}}=\sqrt[3]{3}$$

Insert the obtained value back in the expression.

$$\sqrt[3]{3}+\sqrt[3]{3}-\sqrt[6]{9}$$

Combine \sqrt[3]{3} and \sqrt[3]{3} to get 2\sqrt[3]{3}.

$$2\sqrt[3]{3}-\sqrt[6]{9}$$

Rewrite \sqrt[6]{9} as \sqrt[6]{3^{2}}. Convert from radical to exponential form and cancel out 2 in the exponent. Convert back to radical form.

$$\sqrt[6]{9}=\sqrt[6]{3^{2}}=3^{\frac{2}{6}}=3^{\frac{1}{3}}=\sqrt[3]{3}$$

Insert the obtained value back in the expression.

$$2\sqrt[3]{3}-\sqrt[3]{3}$$

Combine 2\sqrt[3]{3} and -\sqrt[3]{3} to get \sqrt[3]{3}.

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