MaxWong

\(\text{A} = 2(1\cdot 1) + 4(16\cdot 1) = 66 \text { square units}\\ \text{B} = 2(8\cdot 2) + 2(8\cdot 1) + 2(2\cdot 1) = 52 \text { square units}\\ |\text{A} - \text{B}| = 66 - 52 = 14\text{ square units}\)

\(\dfrac{3}4 x + \dfrac{5}8 = 4x\\ 6x + 5 = 32x\\ 26x = 5\\ x = \dfrac{5}{26}\)

\(\phantom{= }\sqrt 5 + \dfrac{1}{\sqrt5} + \sqrt 7 + \dfrac{1}{\sqrt7}\\ = \dfrac{6\sqrt5}{5} + \dfrac{8\sqrt7}{7}\\ = \dfrac{42\sqrt5}{35} + \dfrac{40\sqrt7}{35}\\ = \dfrac{42\sqrt5 + 40\sqrt7}{35}\\ a + b + c = 42 + 40 + 35 = 117\)

\(\dfrac{1}{r^3 + 7} - 7 = \dfrac{-r^3}{r^3 + 7}\\ 1 - 7(r^3 + 7) = -r^3, r \neq \sqrt[3]{7}\\ 1 - 7r^3 - 49 = -r^3, r \neq \sqrt[3]{7}\\ 6r^3 = -48, r \neq \sqrt[3]{7}\\ r^3 = -8\\ r = -2, -2\omega, -2\omega^2\text{ where }\omega\text{ and }\omega^2\text{ denotes the cube roots of unity.}\)

   123456789²  -  2 * 123456789 * 123456787  +  123456787²

= (123456789 - 123456787)²

= 2²

= 4

7)

\(\quad 2231_4\\ = 10101101_2\)

6)

\(35 = 2^5 + 2^1 + 2^0\\ 35 = 100011_2\)

5)

\(\quad 999_{16}\\ = 9(16^2 + 16 + 1)\\ = 2457\)

4)

\(\quad 6032_8\\ = 6\cdot 8^3 + 3\cdot 8 + 2\\ = 3098\)

3) 

\(\quad0.04\overline{55}\\ = 0.04 + 0.00\overline{5}\\ = \dfrac{4}{100} + \dfrac{0.\overline{5}}{100}\\ = \dfrac{4}{100} + \dfrac{5}{900}\\ = \dfrac{41}{900}\)