Question

Simplify the expression by first transforming the radical to exponential form. Leave the answer in exact form as a radical or a power, not as a decimal approximation.

Answer 1

`\textsf{Radical form}: \quad \sqrt[4]{2}\\\\\textsf{Exponent form}: \quad 2^{(1)/(4)}`

Explanation:

Given expression:

`√(8) / \sqrt[4]{32}`

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

`\implies 8^{(1)/(2)} / 32^{(1)/(4)}`

Rewrite 8 as 2³ and 32 as 2⁵:

`\implies (2^3)^{(1)/(2)} / (2^5)^{(1)/(4)}`

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

`\implies 2^{(3)/(2)} / 2^{(5)/(4)}`

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

`\implies 2^{(3)/(2)-(5)/(4)}`

`\implies 2^{(6)/(4)-(5)/(4)}`

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

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

`\implies \sqrt[4]{2}`