Question

How do I solve this? I just need to know the steps to solve.

Answer 1

`\frac{√(2)\sqrt[4]{3}}{3}`

Explanation:

Given radical expression:

`\frac{\sqrt[4]{4}}{\sqrt[4]{27}}`

Rewrite 4 as 2² and 27 as 3³:

`\implies \frac{\sqrt[4]{2^2}}{\sqrt[4]{3^3}}`

`\textsf{Apply the exponent rule:} \quad \sqrt[n]{a}=a^{(1)/(n)}`

`\implies \frac{\left(2^2\right)^{(1)/(4)}}{\left(3^3\right)^{(1)/(4)}}`

`\textsf{Apply the exponent rule:} \quad (a^b)^c=a^(bc)`

`\implies \frac{2^{(2)/(4)}}{3^{(3)/(4)}}`

Simplify the numerator:

`\implies \frac{2^{(1)/(2)}}{3^{(3)/(4)}}`

`\implies \frac{√(2)}{3^{(3)/(4)}}`

Multiply the numerator and denominator by
`3^{(1)/(4)}`:

`\implies \frac{√(2)\cdot 3^{(1)/(4)}}{3^{(3)/(4)}\cdot 3^{(1)/(4)}}`

`\textsf{Apply the exponent rule:} \quad a^b \cdot a^c=a^(b+c)`

`\implies \frac{√(2)\cdot 3^{(1)/(4)}}{3^{(3)/(4)+(1)/(4)}}`

`\implies \frac{√(2)\cdot 3^{(1)/(4)}}{3^1}`

`\implies \frac{√(2)\cdot 3^{(1)/(4)}}{3}`

`\textsf{Apply the exponent rule:} \quad a^{(1)/(n)}=\sqrt[n]{a}`

`\implies \frac{√(2)\sqrt[4]{3}}{3}`