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The expression
[tex]{x}^{7} - 128[/tex]
is factorable by the factor theorem: since x = 2 is a root (a number which yields 0 when substituted), x - 2 is a factor.
However, dividing x⁷ - 128 by x - 2 gives a quotient which is a sixth-degree polynomial. So, the easiest way to solve for x is by setting x⁷ - 128 equal to 0.
[tex] {x}^{7} - 128 = 0[/tex]
First, we isolate the variable by adding 128 to both sides:
[tex] {x}^{7} = 128[/tex]
Notice that 128 is a power of 2. Precisely, the seventh power of 2. So, we may rewrite 128 as 2⁷.
[tex] {x}^{7} = {2}^{7} [/tex]
By laws of exponents, since the powers are equal, the bases must be equal.
Therefore, x = 2.