1. **Problem:** Simplify the expression $$\frac{(a^{-2} b^{3})^{2}}{a^{2} b^{-1}}$$. 2. **Formula and rules:** When raising a power to another power, multiply exponents: $$(x^m)^n = x^{mn}$$. When dividing like bases, subtract exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. 3. **Step-by-step simplification:** - First, simplify the numerator: $$(a^{-2} b^{3})^{2} = a^{-2 \times 2} b^{3 \times 2} = a^{-4} b^{6}$$. - Now the expression is $$\frac{a^{-4} b^{6}}{a^{2} b^{-1}}$$. - Apply division rule for $a$: $$a^{-4 - 2} = a^{-6}$$. - Apply division rule for $b$: $$b^{6 - (-1)} = b^{6 + 1} = b^{7}$$. 4. **Final simplified expression:** $$a^{-6} b^{7}$$. --- 1. **Problem:** Simplify $$\left(a + \frac{1}{a}\right)^2 - \left(a - \frac{1}{a}\right)^2$$. 2. **Formula and rules:** Use the identity for difference of squares: $$x^2 - y^2 = (x - y)(x + y)$$. 3. **Step-by-step simplification:** - Let $$x = a + \frac{1}{a}$$ and $$y = a - \frac{1}{a}$$. - Then the expression is $$x^2 - y^2 = (x - y)(x + y)$$. - Calculate $$x - y = \left(a + \frac{1}{a}\right) - \left(a - \frac{1}{a}\right) = a + \frac{1}{a} - a + \frac{1}{a} = 2 \times \frac{1}{a} = \frac{2}{a}$$. - Calculate $$x + y = \left(a + \frac{1}{a}\right) + \left(a - \frac{1}{a}\right) = a + \frac{1}{a} + a - \frac{1}{a} = 2a$$. - Multiply: $$\frac{2}{a} \times 2a = 4$$. 4. **Final simplified result:** $$4$$. --- 1. **Problem:** Simplify $$ (7x^2 + 3x + 5) - (-3x^2 + 7x + y - 9) $$. 2. **Formula and rules:** When subtracting polynomials, distribute the minus sign and combine like terms. 3. **Step-by-step simplification:** - Distribute minus: $$7x^2 + 3x + 5 + 3x^2 - 7x - y + 9$$. - Combine like terms: - $$7x^2 + 3x^2 = 10x^2$$ - $$3x - 7x = -4x$$ - Constants: $$5 + 9 = 14$$ - Variable $$-y$$ remains as is. 4. **Final simplified expression:** $$10x^2 - 4x - y + 14$$.