1. Simplify (a) $\frac{9}{11} + \frac{10}{22}$ Step 1: Find common denominator. The denominators are 11 and 22. The least common denominator (LCD) is 22. Step 2: Convert $\frac{9}{11}$ to denominator 22: $$\frac{9}{11} = \frac{9 \times 2}{11 \times 2} = \frac{18}{22}$$ Step 3: Add the fractions with common denominator: $$\frac{18}{22} + \frac{10}{22} = \frac{18 + 10}{22} = \frac{28}{22}$$ Step 4: Simplify the fraction by dividing numerator and denominator by 2: $$\frac{28 \div 2}{22 \div 2} = \frac{14}{11}$$ 2. Simplify (b) $\frac{5}{35} \times \frac{6}{42}$ Step 1: Simplify each fraction first. $$\frac{5}{35} = \frac{1}{7} \text{ (divide numerator and denominator by 5)}$$ $$\frac{6}{42} = \frac{1}{7} \text{ (divide numerator and denominator by 6)}$$ Step 2: Multiply the simplified fractions: $$\frac{1}{7} \times \frac{1}{7} = \frac{1}{49}$$ 3. Write a Pythagorean triplet with one member 14. Step 1: Recall the formula for Pythagorean triplets where one leg is a: $$ (a, \frac{m^2 - n^2}{2}, \frac{m^2 + n^2}{2}) $$ Step 2: Instead, try simple known triplet multiples. Since 14 is even, the triplet including 14 can be formed by scaling (7, 24, 25) by 2: $$ (14, 48, 50) $$ Verify: $$ 14^2 + 48^2 = 196 + 2304 = 2500 $$ $$ 50^2 = 2500 $$ Thus, the triplet is $(14, 48, 50)$. 4. Find square root of 1024 using prime factorization. Step 1: Prime factorize 1024. Note: $1024 = 2^{10}$ since $2^{10} = 1024$ Step 2: Square root of $1024$ is: $$ \sqrt{2^{10}} = 2^{10/2} = 2^5 = 32 $$ 5. Is 13824 a perfect cube? Step 1: Prime factorize 13824. Divide by 2 repeatedly: $13824 \div 2 = 6912$ $6912 \div 2 = 3456$ $3456 \div 2 = 1728$ $1728 \div 2 = 864$ $864 \div 2 = 432$ $432 \div 2 = 216$ $216 \div 2 = 108$ $108 \div 2 = 54$ $54 \div 2 = 27$ $27 \div 3 = 9$ $9 \div 3 = 3$ $3 \div 3 = 1$ Counting powers: Number of 2's = 9 Number of 3's = 3 Step 2: Check if powers of prime factors are multiples of 3 (for perfect cube): $9$ (for 2) is divisible by 3 $3$ (for 3) is divisible by 3 Step 3: Since all prime powers are multiples of 3, 13824 is a perfect cube. 6. Find the exterior angle of a polygon with 9 sides. Step 1: Recall the formula for each exterior angle of a regular polygon: $$ \text{exterior angle} = \frac{360^{\circ}}{n} $$ where $n$ is the number of sides. Step 2: For $n=9$, $$ \text{exterior angle} = \frac{360}{9} = 40^{\circ} $$ 7. Find the value of $X$ in the figure where angles are 92°, 28°, 38°, and $X$ as described. Step 1: Angles on a straight line sum to 180°. Given angles 92° and 28° are adjacent along a straight line with $X$ and 38°. Step 2: Sum relationship: $$ 92 + 28 = X + 38 $$ $$ 120 = X + 38 $$ Step 3: Solve for $X$: $$ X = 120 - 38 = 82^{\circ} $$ Final answers: (a) $\frac{14}{11}$ (b) $\frac{1}{49}$ Pythagorean triplet with 14: $(14,48,50)$ Square root of 1024: 32 13824 is a perfect cube: Yes Exterior angle (9 sides): $40^{\circ}$ Value of $X$: $82^{\circ}$