Answer:
[tex]\dfrac{2}{9}[/tex]
Step-by-step explanation:
Given expression:
[tex]\sqrt[3]{\dfrac{8}{729}}[/tex]
[tex]\textsf{Apply exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:[/tex]
[tex]\implies \left(\dfrac{8}{729}\right)^{\frac{1}{3}}[/tex]
[tex]\textsf{Apply exponent rule} \quad \left(\dfrac{a}{b}\right)^c=\dfrac{a^c}{b^c}:[/tex]
[tex]\implies \dfrac{8^{\frac{1}{3}}}{729^{\frac{1}{3}}}[/tex]
Rewrite 8 as (2 · 2 · 2) and rewrite 729 as (9 · 9 · 9):
[tex]\implies \dfrac{(2 \cdot 2 \cdot 2)^{\frac{1}{3}}}{(9 \cdot 9 \cdot 9)^{\frac{1}{3}}}[/tex]
[tex]\implies \dfrac{(2^3)^{\frac{1}{3}}}{(9^3)^{\frac{1}{3}}}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies \dfrac{2^{(3 \cdot \frac{1}{3})}}{9^{(3 \cdot \frac{1}{3})}}[/tex]
[tex]\implies \dfrac{2^1}{9^1}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^1=a:[/tex]
[tex]\implies \dfrac{2}{9}[/tex]
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