Answer
$9$
Work Step by Step
${{a}^{3}}-26{{\left( a \right)}^{\frac{3}{2}}}-27=0$.
Let $u={{\left( a \right)}^{\frac{3}{2}}}$
${{a}^{3}}-26{{\left( a \right)}^{\frac{3}{2}}}-27=0$.
$\begin{align}
& {{u}^{2}}-26u-27=0 \\
& {{u}^{2}}-27u+u-27=0 \\
& u\left( u-27 \right)+1\left( u-27 \right)=0 \\
& \left( u-27 \right)\left( u+1 \right)=0
\end{align}$
Therefore,
$u=27$ or $u=-1$.
Now, replacing u with ${{\left( a \right)}^{\frac{3}{2}}}$:
$\begin{align}
& {{\left( a \right)}^{\frac{3}{2}}}=27 \\
& a={{\left( 27 \right)}^{\frac{2}{3}}} \\
& a={{\left( {{\left( 3 \right)}^{3}} \right)}^{\frac{2}{3}}} \\
& a=9
\end{align}$
Therefore, the value of a is $9$.
Now, replacing u with ${{\left( a \right)}^{\frac{3}{2}}}$:
${{\left( a \right)}^{\frac{3}{2}}}=-1$, has no solution.
Thus, the value of the expression ${{a}^{3}}-26{{\left( a \right)}^{\frac{3}{2}}}-27=0$ is $9$.
