Factorise the following as far as possible in prime numbers: \( x^{16}-30 x^{8} \sqrt{y}+225 y \) Simplify the following expressions completely: \( \frac{1-\frac{1}{x+1}}{1+\frac{1}{x^{2}-1}} \)
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To begin the factorization of \( x^{16} - 30 x^{8} \sqrt{y} + 225 y \), we recognize this as a quadratic in terms of \( x^8 \). Letting \( z = x^8 \), the expression rewrites as \( z^2 - 30z\sqrt{y} + 225y \). We can factor this expression using the quadratic formula or by recognizing that it is a perfect square: \( (z - 15\sqrt{y})^2 \). Hence, \[ x^{16} - 30 x^{8} \sqrt{y} + 225 y = (x^8 - 15\sqrt{y})^2. \] Next, simplifying \( \frac{1 - \frac{1}{x+1}}{1 + \frac{1}{x^2 - 1}} \): First, simplify the numerator: \[ 1 - \frac{1}{x+1} = \frac{(x+1) - 1}{x+1} = \frac{x}{x+1}. \] For the denominator: \[ 1 + \frac{1}{x^2 - 1} = \frac{(x^2 - 1) + 1}{x^2 - 1} = \frac{x^2}{x^2 - 1}. \] Combining these gives: \[ \frac{\frac{x}{x+1}}{\frac{x^2}{x^2 - 1}} = \frac{x \cdot (x^2 - 1)}{x^2 \cdot (x + 1)} = \frac{x(x^2 - 1)}{x^2(x + 1)}. \] Thus, the final simplified expression is: \[ \frac{x(x - 1)(x + 1)}{x^2(x + 1)} = \frac{x(x - 1)}{x^2} = \frac{x - 1}{x}. \]
