(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )(1+ $\frac{x}{2}$ )

asked by guest
on Nov 19, 2024 at 12:16 pm



You asked:

Evaluate the expression: \(\left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right)\)

MathBot Answer:

Evaluated



\(\displaystyle \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) = \left(\frac{x}{2} + 1\right)^{10} \)

Expanded

\[\left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) = \frac{x^{10}}{1024} + \frac{5 x^{9}}{256} + \frac{45 x^{8}}{256} + \frac{15 x^{7}}{16} + \frac{105 x^{6}}{32} + \frac{63 x^{5}}{8} + \frac{105 x^{4}}{8} + 15 x^{3} + \frac{45 x^{2}}{4} + 5 x + 1\]

Factored

\[\left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) \cdot \left(1 + \frac{x}{2}\right) = \frac{\left(x + 2\right)^{10}}{1024}\]

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