Đáp án:

$\begin{array}{l}
a){x^{15}} = x.1\\
 \Rightarrow {x^{15}} - x = 0\\
 \Rightarrow x.\left( {{x^{14}} - 1} \right) = 0\\
 \Rightarrow \left[ \begin{array}{l}
x = 0\\
{x^{14}} = 1
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
x = 0\\
x = 1\\
x =  - 1
\end{array} \right.\\
\text{Vậy}\,x = 0;x = 1;x =  - 1\\
c){\left( {2x + 1} \right)^3} = 125\\
 \Rightarrow {\left( {2x + 1} \right)^3} = {5^3}\\
 \Rightarrow 2x + 1 = 5\\
 \Rightarrow 2x = 4\\
 \Rightarrow x = 2\\
\text{Vậy}\,x = 2\\
d){\left( {x + 5} \right)^4} = {\left( {x + 5} \right)^6}\\
 \Rightarrow {\left( {x + 5} \right)^4}.\left[ {{{\left( {x + 5} \right)}^2} - 1} \right] = 0\\
 \Rightarrow \left[ \begin{array}{l}
x + 5 = 0\\
{\left( {x + 5} \right)^2} = 1
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
x =  - 5\\
x + 5 = 1\\
x + 5 =  - 1
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
x =  - 5\\
x =  - 4\\
x =  - 6
\end{array} \right.\\
\text{Vậy}\,x =  - 6;x =  - 5;x =  - 4\\
e){x^{10}} = x\\
 \Rightarrow {x^{10}} - x = 0\\
 \Rightarrow x\left( {{x^9} - 1} \right) = 0\\
 \Rightarrow \left[ \begin{array}{l}
x = 0\\
{x^9} = 1
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
x = 0\\
x = 1
\end{array} \right.\\
\text{Vậy}\,x = 0;x = 1\\
f){\left( {2x - 15} \right)^5} = {\left( {2x - 15} \right)^3}\\
 \Rightarrow {\left( {2x - 15} \right)^3}.\left( {{{\left( {2x - 15} \right)}^2} - 1} \right) = 0\\
 \Rightarrow \left[ \begin{array}{l}
\left( {2x - 15} \right) = 0\\
{\left( {2x - 15} \right)^2} = 1
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
2x = 15\\
2x - 15 = 1\\
2x - 15 =  - 1
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
x = \dfrac{{15}}{2}\\
x = 8\\
x = 7
\end{array} \right.\\
\text{Vậy}\,x = 7;x = \dfrac{{15}}{2};x = 8
\end{array}$