Answer
(a) $6\sqrt{2}$
(b) $48a^5b^7$
(c) $\frac{x}{9y^7}$
(d) $\frac{x + 2}{x - 2}, x\not=-1, x\not=2$
(e) $-\frac{x - 4}{(x - 2)(x + 2)}$
(f) $-y-x$
Work Step by Step
(a) $$\sqrt{200} - \sqrt{32} = \sqrt{100 \times 2} - \sqrt{16 \times 2} = 10\sqrt{2} - 4\sqrt{2} = 6\sqrt{2}$$
(b) $$(3a^3b^3)(4ab^2)^2 = 3a^3b^3 \cdot 16a^2b^4 = 48a^{3+2}b^{3+4} = 48a^5b^7$$
(c) $$\left( \frac{3x^{3/2}y^3}{x^2y^{-1/2}} \right)^{-2} = \left( \frac{3x^{3/2}}{x^2} \cdot \frac{y^3}{y^{-1/2}} \right)^{-2} = \left( 3x^{3/2-2}y^{3+1/2} \right)^{-2} = \left( 3x^{-1/2}y^{7/2} \right)^{-2} = \left( 3 \cdot x^{-1/2} \cdot y^{7/2} \right)^{-2} =\frac{1}{9} x^{1}y^{-7} = \frac{x}{9y^7} $$
(d) $$\frac{x^2 + 3x + 2}{x^2 - x - 2} = \frac{(x + 2)(x + 1)}{(x + 1)(x - 2)} = \frac{x + 2}{x - 2} \text{ (assuming
x≠−
1
and
x
≠
2
)}$$
(e) $$\frac{x^2}{x^2 - 4} - \frac{x + 1}{x + 2} = \frac{x^2}{(x - 2)(x + 2)} - \frac{x + 1}{x + 2} = \frac{x^2 - (x + 1)(x - 2)}{(x - 2)(x + 2)} = \frac{x^2 - x^2 + 2 - x + 2}{(x - 2)(x + 2)} = \frac{4 - x}{(x - 2)(x + 2)} = \frac{-(x - 4)}{(x - 2)(x + 2)} = -\frac{(x - 4)}{(x - 2)(x + 2)}$$
(f) $$\frac{\frac{y}{x} - \frac{x}{y}}{\frac{1}{y} - \frac{1}{x}} = \frac{\frac{y^2 - x^2}{xy}}{\frac{x - y}{xy}} = \frac{y^2 - x^2}{x - y} = \frac{(y - x)(y + x)}{x - y} = \frac{(y + x)(y - x)}{-(y - x)} = -(y + x) = -y - x$$
