What is \sqrt[3]{\sqrt[4]{27a^3}} ?

The solution to \sqrt[3]{\sqrt[4]{27a^3}} is \sqrt[4]{3}\sqrt[12]{a^3}

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simplify \sqrt[3]{\sqrt[4]{27a^3}}

simplify

3 4 27 a^{3}

43 12 a^{3}

Solution steps

3 4 27 a^{3}

Apply radical rule:

n a = a^{\frac{1}{n}}

= ((27 a^{3})^{\frac{1}{4}})^{\frac{1}{3}}

Apply exponent rule:

(a^{b})^{c} = a^{b c},

assuming

a \geq 0

= (27 a^{3})^{\frac{1}{4} \cdot \frac{1}{3}}

\frac{1}{4} \cdot \frac{1}{3} = \frac{1}{12}

= (27 a^{3})^{\frac{1}{12}}

a^{\frac{1}{n}} = n a

= 12 27 a^{3}

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

= 1227 12 a^{3}

1227 = 43

= 43 12 a^{3}

35

5 y \cdot 3 y^{2}

(14)^{2}

7 5^{2}

3^{2} e^{- 3}

What is \sqrt[3]{\sqrt[4]{27a^3}} ?
The solution to \sqrt[3]{\sqrt[4]{27a^3}} is \sqrt[4]{3}\sqrt[12]{a^3}