Answer:
[tex]27^{\frac{1}{3}} \quad 125^{\frac{2}{3}} \quad 9^{\frac{3}{2}}[/tex]
Step-by-step explanation:
Rewrite 9 as 3²:
[tex]\implies 9^{\frac{3}{2}}=\left(3^2\right)^{\frac{3}{2}}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 3^{(2 \cdot \frac{3}{2})}=3^3[/tex]
Therefore:
[tex]\implies 3^3= 3 \cdot 3 \cdot 3=27[/tex]
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Rewrite 27 as 3³:
[tex]\implies 27^{\frac{1}{3}}=\left(3^3\right)^{\frac{1}{3}}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 3^{(3 \cdot \frac{1}{3})}=3^1[/tex]
Therefore:
[tex]\implies 3^1=3[/tex]
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Rewrite 125 as 5³:
[tex]\implies 125^{\frac{2}{3}}=\left(5^3\right)^{\frac{2}{3}}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 5^{(3 \cdot \frac{2}{3})}=5^2[/tex]
Therefore:
[tex]\implies 5^2=5 \cdot 5=25[/tex]
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Solution
In order, from smallest to largest:
[tex]27^{\frac{1}{3}} \quad 125^{\frac{2}{3}} \quad 9^{\frac{3}{2}}[/tex]
