Simplify sqrt(8).

Fresh day More, let's solve.

a) To simplify the expression, we first simplify $\sqrt{8}$ and then rationalize the denominator.

Step 1: Simplify $\sqrt{8}$. $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$

Step 2: Substitute the simplified term into the expression. $\frac{2\sqrt{2}-2}{4-\sqrt{2}}$

Step 3: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $4+\sqrt{2}$. $\frac{2\sqrt{2}-2}{4-\sqrt{2}} \times \frac{4+\sqrt{2}}{4+\sqrt{2}}$

Step 4: Expand the numerator. $(2\sqrt{2}-2)(4+\sqrt{2}) = 2\sqrt{2}(4) + 2\sqrt{2}(\sqrt{2}) - 2(4) - 2(\sqrt{2})$ $= 8\sqrt{2} + 2(2) - 8 - 2\sqrt{2}$ $= 8\sqrt{2} + 4 - 8 - 2\sqrt{2}$ $= (8\sqrt{2} - 2\sqrt{2}) + (4 - 8)$ $= 6\sqrt{2} - 4$

Step 5: Expand the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$. $(4-\sqrt{2})(4+\sqrt{2}) = 4^2 - (\sqrt{2})^2$ $= 16 - 2$ $= 14$

Step 6: Combine the simplified numerator and denominator. $\frac{6\sqrt{2}-4}{14}$

Step 7: Simplify the fraction by dividing both the numerator and the denominator by their greatest common factor, which is 2. $\frac{2(3\sqrt{2}-2)}{2(7)}$ $= \frac{3\sqrt{2}-2}{7}$

The simplified expression is $\boxed{\frac{3\sqrt{2}-2}{7}}$.

b) To express the given expression as a single fraction, we first factor the denominator of the first term.

Step 1: Factor the denominator $y^2-y-2$. We look for two numbers that multiply to -2 and add to -1. These numbers are -2 and 1. $y^2-y-2 = (y-2)(y+1)$

Step 2: Rewrite the expression with the factored denominator. $\frac{1}{(y-2)(y+1)} - \frac{1}{y+1}$

Step 3: Find a common denominator, which is $(y-2)(y+1)$. To get this common denominator for the second term, multiply its numerator and denominator by $(y-2)$. $\frac{1}{y+1} = \frac{1}{y+1} \times \frac{y-2}{y-2} = \frac{y-2}{(y-2)(y+1)}$

Step 4: Perform the subtraction. $\frac{1}{(y-2)(y+1)} - \frac{y-2}{(y-2)(y+1)}$ $= \frac{1 - (y-2)}{(y-2)(y+1)}$

Step 5: Simplify the numerator. $1 - (y-2) = 1 - y + 2 = 3 - y$

Step 6: Write the final single fraction. $\frac{3-y}{(y-2)(y+1)}$

The expression as a single fraction is $\boxed{\frac{3-y}{(y-2)(y+1)}}$.

a) Let the first term of the Arithmetic Progression (AP) be $a$ and the common difference be $d$. The first three terms are $a$, $a+d$, and $a+2d$.

Step 1: Use the information about the average of the first three terms. The average of the first three terms is 56. $\frac{a + (a+d) + (a+2d)}{3} = 56$ $\frac{3a + 3d}{3} = 56$ $a + d = 56 \quad (*)$

Step 2: Use the information about the sum of the first two terms. The sum of the first two terms is 96. $a + (a+d) = 96$ $2a + d = 96 \quad (**)$

Step 3: Solve the system of linear equations for $a$ and $d$. Subtract equation $(*)$ from equation $(**)$: $(2a + d) - (a + d) = 96 - 56$ $a = 40$

Step 4: Substitute the value of $a$ into equation $(*)$ to find $d$. $40 + d = 56$ $d = 56 - 40$ $d = 16$

The first term is $\boxed{40}$ and the common difference is $\boxed{16}$.

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