\frac{x\left(\left(x\cdot t+\left(x-1\right)\right)\cdot\left(t+1\right)^{\frac{1}{x}}-\left(x\cdot\left(t+1\right)\right)\cdot t^{\frac{1}{x}}\right)}{\left(x-1\right)\cdot t\cdot(t+1)}

asked by guest
on Jan 23, 2025 at 4:46 pm



You asked:

Evaluate the expression: \(\frac{x \left(\left(x t + x - 1\right) {\left( t + 1 \right)}^{\frac{1}{x}} - x \left(t + 1\right) \cdot {t}^{\frac{1}{x}}\right)}{\left(x - 1\right) \cdot t \left(t + 1\right)}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{x \left(\left(x t + x - 1\right) {\left( t + 1 \right)}^{\frac{1}{x}} - x \left(t + 1\right) \cdot {t}^{\frac{1}{x}}\right)}{\left(x - 1\right) \cdot t \left(t + 1\right)} = \frac{x \left(- t^{\frac{1}{x}} x \left(t + 1\right) + \left(t + 1\right)^{\frac{1}{x}} \left(t x + x - 1\right)\right)}{t \left(t + 1\right) \left(x - 1\right)} \)

Expanded

\[\frac{x \left(\left(x t + x - 1\right) {\left( t + 1 \right)}^{\frac{1}{x}} - x \left(t + 1\right) \cdot {t}^{\frac{1}{x}}\right)}{\left(x - 1\right) \cdot t \left(t + 1\right)} = - \frac{t^{\frac{1}{x}} x^{2} t}{t^{2} x - t^{2} + t x - t} - \frac{t^{\frac{1}{x}} x^{2}}{t^{2} x - t^{2} + t x - t} + \frac{x^{2} t \left(t + 1\right)^{\frac{1}{x}}}{t^{2} x - t^{2} + t x - t} + \frac{x^{2} \left(t + 1\right)^{\frac{1}{x}}}{t^{2} x - t^{2} + t x - t} - \frac{x \left(t + 1\right)^{\frac{1}{x}}}{t^{2} x - t^{2} + t x - t}\]

Factored

\[\frac{x \left(\left(x t + x - 1\right) {\left( t + 1 \right)}^{\frac{1}{x}} - x \left(t + 1\right) \cdot {t}^{\frac{1}{x}}\right)}{\left(x - 1\right) \cdot t \left(t + 1\right)} = \frac{x \left(- t^{\frac{1}{x}} t x - t^{\frac{1}{x}} x + t x \left(t + 1\right)^{\frac{1}{x}} + x \left(t + 1\right)^{\frac{1}{x}} - \left(t + 1\right)^{\frac{1}{x}}\right)}{t \left(t + 1\right) \left(x - 1\right)}\]

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