Answer:
[tex]x = 17[/tex].
Step-by-step explanation:
Simplify the given equation:
[tex]2\, (x - 1)^{3/4} = 16[/tex].
[tex](x - 1)^{3/4} = 8[/tex].
[tex]((x - 1)^{1/4})^{3} = 8[/tex].
Take the cubic root of both sides to eliminate the numerator of the exponent. Only roots of even degree could introduce alternative solutions. Since the degree of this root is an odd number, this step would not create extraneous solutions.
[tex]\sqrt[3]{((x - 1)^{1/4})^{3}} = \sqrt[3]{8}[/tex].
[tex](x - 1)^{1/4} = 2[/tex].
Raise both sides to the fourth power to eliminate the denominator of the exponent:
[tex](x - 1) = 2^{4}[/tex].
[tex]x - 1 = 16[/tex].
[tex]x = 17[/tex].
Substitute the solution [tex]x = 17[/tex] into the equation:
[tex]\begin{aligned}& 2\, (17 - 1)^{3/4} \\ =\; & 2\times (16^{1/4})^{3} \\ =\; & 2 \times 2^{3} \\ =\; & 16\end{aligned}[/tex].
Thus, [tex]x = 17[/tex] isn't an extraneous solution since setting [tex]x[/tex] to [tex]17[/tex] satisfies the original equation.
