Answer:
[tex]\textsf{Radical form}: \quad \sqrt[4]{2}\\\\\textsf{Exponent form}: \quad 2^{\frac{1}{4}}[/tex]
Step-by-step explanation:
Given expression:
[tex]\sqrt{8} \div \sqrt[4]{32}[/tex]
[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:[/tex]
[tex]\implies 8^{\frac{1}{2}} \div 32^{\frac{1}{4}}[/tex]
Rewrite 8 as 2³ and 32 as 2⁵:
[tex]\implies (2^3)^{\frac{1}{2}} \div (2^5)^{\frac{1}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 2^{\frac{3}{2}} \div 2^{\frac{5}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad a^b \div a^c=a^{b-c}:[/tex]
[tex]\implies 2^{\frac{3}{2}-\frac{5}{4}}[/tex]
[tex]\implies 2^{\frac{6}{4}-\frac{5}{4}}[/tex]
[tex]\implies 2^{\frac{1}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:[/tex]
[tex]\implies \sqrt[4]{2}[/tex]
