Answer:
[tex]p= \dfrac{1}{2}[/tex]
Step-by-step explanation:
Given equation:
[tex]p^3=\dfrac{1}{8}[/tex]
Cube root both sides:
[tex]\implies \sqrt[3]{p^3}= \sqrt[3]{\dfrac{1}{8}}[/tex]
[tex]\implies p= \sqrt[3]{\dfrac{1}{8}}[/tex]
[tex]\textsf{Apply exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:[/tex]
[tex]\implies p= \left(\dfrac{1}{8}\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 p= \dfrac{1^{\frac{1}{3}}}{8^{\frac{1}{3}}}[/tex]
[tex]\textsf{Apply exponent rule} \quad 1^a=1:[/tex]
[tex]\implies p= \dfrac{1}{8^{\frac{1}{3}}}[/tex]
Rewrite 8 as 2³:
[tex]\implies p= \dfrac{1}{(2^3)^{\frac{1}{3}}}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies p= \dfrac{1}{2^{(3 \cdot \frac{1}{3})}}[/tex]
Simplify:
[tex]\implies p= \dfrac{1}{2^{\frac{3}{3}}}[/tex]
[tex]\implies p= \dfrac{1}{2^{1}}[/tex]
[tex]\implies p= \dfrac{1}{2}[/tex]
