1a. Rationalize the following expressions: 1. $$\frac{2}{3-\sqrt{2}} + \frac{1}{3+\sqrt{2}}$$ 2. $$\frac{3\sqrt{5} + 2\sqrt{3}}{2\sqrt{5} - 3\sqrt{3}}$$ 3. $$\frac{\sqrt{6} - 2\sqrt{3}}{\sqrt{6} + \sqrt{3}}$$ 4. $$\frac{10}{4\sqrt{18} - 3\sqrt{48}}$$ 5. $$\frac{\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} - 2\sqrt{2}}$$ **Step 1:** Use the conjugate to rationalize denominators of fractions. **Step 2:** Multiply numerator and denominator by the conjugate of the denominator. **Step 3:** Simplify radicals and expressions. --- 1b. Evaluate without tables: 1. $$\frac{\sqrt{50} - \sqrt{98}}{\sqrt{32}}$$ 2. $$\frac{1}{\sqrt{3} - 2} - \frac{1}{\sqrt{3} + 2}$$ 3. $$\frac{10\sqrt{32}}{\sqrt{18} - \sqrt{2}}$$ **Step 1:** Simplify radicals. **Step 2:** Rationalize denominators if needed. **Step 3:** Perform arithmetic operations. --- 2a. Given matrices: $$A = \begin{bmatrix}4 & 6 \\ -2 & 7 \\ 5 & -3\end{bmatrix}, B = \begin{bmatrix}-5 & 8 \\ -4 & 3 \\ 2 & -7\end{bmatrix}, C = \begin{bmatrix}0 & 6 \\ -7 & 2 \\ 5 & 0\end{bmatrix}$$ Evaluate: (a) $$A + B$$ (b) $$B + C$$ (c) $$A - B + C$$ (d) $$B - A + C$$ **Step 1:** Add or subtract corresponding elements. --- 2b. Given: $$A = \begin{bmatrix}2 & 5 \\ 3 & -4\end{bmatrix}, B = \begin{bmatrix}3 & 6 \\ 2 & 0\end{bmatrix}$$ Show that $$A(2B) = 2(AB)$$ and calculate $$A^2 - B^2$$. **Step 1:** Calculate $$2B$$. **Step 2:** Calculate $$A(2B)$$ and $$2(AB)$$. **Step 3:** Verify equality. **Step 4:** Calculate $$A^2 = A \times A$$ and $$B^2 = B \times B$$. **Step 5:** Subtract $$B^2$$ from $$A^2$$. --- 2c. Evaluate determinant of $$M = \begin{bmatrix}3 & 9 & 5 \\ 6 & 0 & 7 \\ 4 & 6 & 5\end{bmatrix}$$ **Step 1:** Use determinant formula for 3x3 matrix: $$|M| = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where $$a,b,c,d,e,f,g,h,i$$ are elements of matrix. --- 3a. Calculate determinants of: 1. $$\begin{bmatrix}7 & 8 & 2 \\ 5 & -1 & 2 \\ 6 & 6 & 3\end{bmatrix}$$ 2. $$\begin{bmatrix}9 & 2 & 5 \\ 6 & -1 & 0 \\ -9 & 7 & 2\end{bmatrix}$$ 3. $$\begin{bmatrix}2 & 4 & 7 \\ 3 & 5 & 6 \\ 2 & 0 & 4\end{bmatrix}$$ 4. $$\begin{bmatrix}2 & 1 & 0 \\ 6 & 3 & 2 \\ 3 & 5 & 5\end{bmatrix}$$ **Step 1:** Use determinant formula for 3x3 matrices. --- 3b. Find inverses of: 1. $$\begin{bmatrix}5 & 3 \\ 2 & 2\end{bmatrix}$$ 2. $$\begin{bmatrix}3 & -2 \\ 7 & -1\end{bmatrix}$$ 3. $$\begin{bmatrix}4 & 7 \\ 3 & 4\end{bmatrix}$$ 4. $$\begin{bmatrix}4 & 5 \\ 2 & 3\end{bmatrix}$$ **Step 1:** Calculate determinant. **Step 2:** Use formula for inverse of 2x2 matrix: $$A^{-1} = \frac{1}{|A|} \begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$$ where $$A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$$. --- 4a. Evaluate without tables or calculator: 1. $$\frac{\log 27}{\log 18 - \log 6}$$ 2. $$\frac{\log 0.4}{\log 0.064}$$ 3. $$\frac{\log 64}{\log \frac{1}{4}}$$ 4. $$\log_5 4 - 2\log_5 \frac{5}{6} + \log_5 \frac{625}{144}$$ 5. $$\log_5 \frac{1}{8} - 2\log_4 1\frac{3}{4} + \log_4 \frac{49}{32}$$ 6. $$2\log_3 6 - \log_3 12 + \log_3 27$$ **Step 1:** Use log properties: $$\log a - \log b = \log \frac{a}{b}$$, $$n \log a = \log a^n$$. **Step 2:** Simplify expressions. --- 4b. Given $$\log 2 = 0.3010$$, $$\log 3 = 0.4771$$, $$\log 7 = 0.8451$$, evaluate: 1. $$\log 0.8$$ 2. $$\log 36$$ 3. $$\log 108$$ 4. $$\log 21$$ **Step 1:** Express numbers in prime factors. **Step 2:** Use log addition and subtraction. --- 5. Simplify: 1. $$\frac{4}{5} \times (4\frac{3}{8} - 2\frac{3}{4})$$ 2. $$\frac{2\frac{3}{4} \div 2\frac{5}{8}}{5 \times 3\frac{1}{7}}$$ 3. $$2\frac{3}{4} - 1\frac{5}{6} + \frac{3\frac{1}{2}}{1\frac{3}{8}}$$ 4. $$6\frac{3}{4} \div \left(2\frac{3}{4} \div 3\frac{2}{3}\right)$$ 5. $$\frac{4\frac{1}{5}}{2\frac{3}{10} - 1\frac{2}{15}}$$ 6. $$3\frac{1}{2} \times 6\frac{3}{4} \div 2\frac{1}{4}$$ **Step 1:** Convert mixed numbers to improper fractions. **Step 2:** Perform arithmetic operations stepwise. **Step 3:** Simplify results.