Solve the Exponential Equation: x^14 - x^7 = 0

Shown below is the given equation:

$x^{14}-x^7=0$

Note that on the left side we are able factor the expression by using a common factor. The largest common factor for the numbers and letters in this case is $x^{7}$given that the seventh power is the lowest power in the equation. Therefore it is included both in the term with the 14th power and in the term with the seventh power. Any power higher than this is not included in the term with the lowest seventh power. Hence this is the term with the highest power that can be factored out as a common factor from all terms for the letters,

Let's continue to factor the equation:

$x^{14}-x^7=0 \\ \downarrow\\ x^{7}(x^7-1)=0$

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore due to the fact that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

$x^{7}=0 \hspace{8pt}\text{/}\sqrt[7]{\hspace{6pt}}\\ \boxed{x=0}$

In solving the above equation, we extracted a 99th root for both sides of the equation.

(In this case, extracting an odd-order root to the right side of the equation yielded one possibility)

Or:

$x^7-1=0\\ x^7=1\hspace{8pt}\text{/}\sqrt[7]{\hspace{6pt}}\\ \boxed{x=1}$

In solving the above equation, we first isolated the variable (because it's possible..) on one side and then extracted a seventh root for both sides of the equation.

(In this case, again, extracting an odd-order root to the right side of the equation yielded one possibility)

Let's summarize the solution of the equation:

$x^{14}-x^7=0 \\ \downarrow\\ x^{7}(x^7-1)=0 \\ \downarrow\\ x^{7}=0 \rightarrow\boxed{ x=0}\\ x^{7}-1=0\rightarrow \boxed{x=1}\\ \downarrow\\ \boxed{x=0,1}$

Therefore the correct answer is answer D.