1. **Problem:** Convert the binary numbers $101_2$ and $111_2$ to decimal and add them. 2. **Formula:** To convert binary to decimal, use $\sum_{i=0}^{n} b_i \times 2^i$ where $b_i$ is the bit at position $i$. 3. **Conversion:** - $101_2 = 1\times2^2 + 0\times2^1 + 1\times2^0 = 4 + 0 + 1 = 5$ - $111_2 = 1\times2^2 + 1\times2^1 + 1\times2^0 = 4 + 2 + 1 = 7$ 4. **Addition:** $5 + 7 = 12$ --- 1. **Problem:** Calculate $(27^{1/3})^2$. 2. **Formula:** $(a^{m})^{n} = a^{m \times n}$. 3. **Calculation:** - $27^{1/3} = 3$ because $3^3 = 27$ - So, $(27^{1/3})^2 = 3^2 = 9$ --- 1. **Problem:** Calculate $36^{1/2} \times 64^{-1/4} \times 5^0$. 2. **Formula:** - $a^{1/2} = \sqrt{a}$ - $a^0 = 1$ - $a^{-b} = \frac{1}{a^b}$ 3. **Calculation:** - $36^{1/2} = 6$ - $64^{-1/4} = \frac{1}{64^{1/4}} = \frac{1}{2^3} = \frac{1}{8}$ since $64 = 2^6$ and $64^{1/4} = 2^{6/4} = 2^{3/2} = 2^{1.5} = 2 \times \sqrt{2} \approx 2.828$, but better to calculate $64^{1/4} = \sqrt{\sqrt{64}} = \sqrt{8} \approx 2.828$, so $64^{-1/4} \approx \frac{1}{2.828} \approx 0.3536$ - $5^0 = 1$ 4. **Multiply:** $6 \times 0.3536 \times 1 \approx 2.1216$ --- 1. **Problem:** Find $27^{1/3}$. 2. **Calculation:** $27^{1/3} = 3$ because $3^3 = 27$ --- 1. **Problem:** Solve $\log_x 81 = 2$ and $\log_x 9 = 2$. 2. **Formula:** $\log_x y = z \Rightarrow x^z = y$ 3. **From $\log_x 81 = 2$:** - $x^2 = 81$ - $x = \pm 9$, but base $x$ must be positive and not 1, so $x=9$ 4. **From $\log_x 9 = 2$:** - $x^2 = 9$ - $x = 3$ or $-3$, but base $x$ must be positive and not 1, so $x=3$ 5. **Note:** The two equations contradict if $x$ is the same base. So no single $x$ satisfies both. --- 1. **Problem:** Find $P \cup Q \cup R$ where $P = \{2, 1, 3, 9, \frac{1}{2}\}$, $Q = \{1, 2 \frac{1}{2}, 3, 7\}$, $R = \{5, 4, 2 \frac{1}{2}\}$. 2. **Union:** Combine all unique elements. 3. **Result:** $\{\frac{1}{2}, 1, 2, 2 \frac{1}{2}, 3, 4, 5, 7, 9\}$ --- 1. **Problem:** Find $P \cap Q \cap R$. 2. **Intersection:** Elements common to all three sets. 3. **Result:** $\{2 \frac{1}{2}\}$ --- 1. **Problem:** Calculate $16 \frac{1}{3} \times 2 \frac{3}{8}$. 2. **Convert to improper fractions:** - $16 \frac{1}{3} = \frac{49}{3}$ - $2 \frac{3}{8} = \frac{19}{8}$ 3. **Multiply:** $$\frac{49}{3} \times \frac{19}{8} = \frac{931}{24}$$ 4. **Convert to mixed number:** - $931 \div 24 = 38$ remainder $19$ - So, $38 \frac{19}{24}$ --- 1. **Problem:** Convert $132_6$ to decimal. 2. **Calculation:** - $1 \times 6^2 + 3 \times 6^1 + 2 \times 6^0 = 36 + 18 + 2 = 56$ --- 1. **Problem:** Convert $223_4$ to decimal. 2. **Calculation:** - $2 \times 4^2 + 2 \times 4^1 + 3 \times 4^0 = 32 + 8 + 3 = 43$ --- 1. **Problem:** Multiply $321_6 \times 25_6$. 2. **Convert to decimal:** - $321_6 = 3 \times 6^2 + 2 \times 6 + 1 = 108 + 12 + 1 = 121$ - $25_6 = 2 \times 6 + 5 = 12 + 5 = 17$ 3. **Multiply:** $121 \times 17 = 2057$ --- 1. **Problem:** Add $2115_7 + 12_7$. 2. **Convert to decimal:** - $2115_7 = 2 \times 7^3 + 1 \times 7^2 + 1 \times 7 + 5 = 686 + 49 + 7 + 5 = 747$ - $12_7 = 1 \times 7 + 2 = 9$ 3. **Add:** $747 + 9 = 756$ --- 1. **Problem:** Calculate $16 \frac{1}{3} \times 4 \frac{1}{2}$. 2. **Convert to improper fractions:** - $16 \frac{1}{3} = \frac{49}{3}$ - $4 \frac{1}{2} = \frac{9}{2}$ 3. **Multiply:** $$\frac{49}{3} \times \frac{9}{2} = \frac{441}{6} = 73 \frac{1}{2}$$ --- 1. **Problem:** Calculate $27 \frac{1}{3}$ (interpreted as $27^{1/3}$). 2. **Calculation:** $27^{1/3} = 3$ --- 1. **Problem:** Find $A \cup B$ where $A = \{m, a, p, e\}$ and $B = \{a, e, i, o, u\}$. 2. **Union:** Combine unique elements. 3. **Result:** $\{m, a, p, e, i, o, u\}$ --- 1. **Problem:** Find $A \cap C$ where $C = \{i, m, n, o, p, q, r, s, t, u\}$. 2. **Intersection:** Elements common to both. 3. **Result:** $\{m, p\}$ --- 1. **Problem:** Find $B \cap C$. 2. **Intersection:** $\{i, o, u\}$ --- 1. **Problem:** Find $A \cap B$. 2. **Intersection:** $\{a, e\}$ --- 1. **Problem:** Evaluate $69.24 \times 8.31$. 2. **Calculation:** $69.24 \times 8.31 = 575.2044$ --- 1. **Problem:** Evaluate $7031 \div 4911$. 2. **Calculation:** $7031 \div 4911 \approx 1.432$