1. Problem: Convert the fractions $\frac{3}{4}$, $\frac{9}{50}$, $\frac{7}{100}$, $\frac{29}{20}$ to decimal form. Step 1: Divide numerator by denominator for each fraction. $\frac{3}{4} = 0.75$ $\frac{9}{50} = 0.18$ $\frac{7}{100} = 0.07$ $\frac{29}{20} = 1.45$ Answer: $0.75$, $0.18$, $0.07$, $1.45$ 2. Problem: Convert decimal numbers $13.5$, $0.05349$, $4.039$ to fractions. Step 1: Write each decimal as a fraction with denominator as power of 10. $13.5 = 13 \frac{5}{10} = \frac{135}{10} = \frac{27}{2}$ after simplification. $0.05349 = \frac{5349}{100000}$ (already in fraction form). $4.039 = 4 + 0.039 = 4 + \frac{39}{1000} = \frac{4000}{1000} + \frac{39}{1000} = \frac{4039}{1000}$. Answer: $\frac{27}{2}$, $\frac{5349}{100000}$, $\frac{4039}{1000}$ 3. Problem: Write fractions $\frac{3}{7}$, $\frac{5}{12}$, $\frac{17}{6}$, $\frac{19}{18}$ as periodic decimals. Step 1: Convert each fraction to decimal and identify repeating parts. $\frac{3}{7} = 0.\overline{428571}$ (pure periodic with period 428571) $\frac{5}{12} = 0.41\overline{6}$ (mixed periodic: 41 non-repeating, 6 repeating) $\frac{17}{6} = 2.8\overline{3}$ (mixed periodic: 8 non-repeating, 3 repeating) $\frac{19}{18} = 1.0\overline{5}$ (mixed periodic: 0 non-repeating, 5 repeating) Answer: $0.\overline{428571}$, $0.41\overline{6}$, $2.8\overline{3}$, $1.0\overline{5}$ 4. Problem: Convert fractions in the 3x3 grid to decimals and classify them into final decimal, pure periodic, or mixed periodic. Step 1: Calculate decimals: $\frac{13}{52} = 0.25$ (final decimal) $\frac{2}{7} = 0.\overline{285714}$ (pure periodic) $\frac{11}{6} = 1.8\overline{3}$ (mixed periodic) $\frac{15}{8} = 1.875$ (final decimal) $\frac{13}{27} = 0.\overline{481}$ (pure periodic) $\frac{19}{15} = 1.2\overline{6}$ (mixed periodic) $\frac{1}{2} = 0.5$ (final decimal) $\frac{2}{11} = 0.\overline{18}$ (pure periodic) $\frac{11}{30} = 0.3\overline{6}$ (mixed periodic) Step 2: Classification: - Konačni decimalni broj (final decimal): $0.25$, $1.875$, $0.5$ - Čisto periodični decimalni broj (pure periodic): $0.\overline{285714}$, $0.\overline{481}$, $0.\overline{18}$ - Mješovito periodični decimalni broj (mixed periodic): $1.8\overline{3}$, $1.2\overline{6}$, $0.3\overline{6}$ 5. Problem: Calculate a) $\left(-\frac{2}{7}\right)^2 \times \left(\frac{5}{4}\right)^2$ b) $\left(-1 \frac{3}{5}\right)^2 : \left(1 \frac{1}{4}\right)^2$ Step 1a: Square each fraction: $\left(-\frac{2}{7}\right)^2 = \frac{4}{49}$ $\left(\frac{5}{4}\right)^2 = \frac{25}{16}$ Multiply: $\frac{4}{49} \times \frac{25}{16} = \frac{100}{784} = \frac{25}{196}$ Step 1b: Convert mixed numbers to improper fractions: $-1 \frac{3}{5} = -\frac{8}{5}$ $1 \frac{1}{4} = \frac{5}{4}$ Square: $\left(-\frac{8}{5}\right)^2 = \frac{64}{25}$ $\left(\frac{5}{4}\right)^2 = \frac{25}{16}$ Divide: $\frac{64}{25} : \frac{25}{16} = \frac{64}{25} \times \frac{16}{25} = \frac{1024}{625}$ Answer: a) $\frac{25}{196}$ b) $\frac{1024}{625}$ 6. Problem: Calculate $\left(-\frac{3}{2}\right)^2 \times \left(2 \frac{2}{3}\right)^2 \times \left(2 \frac{1}{4}\right)^2 \times \left(-\frac{2}{3}\right)^2$ Step 1: Convert mixed numbers: $2 \frac{2}{3} = \frac{8}{3}$ $2 \frac{1}{4} = \frac{9}{4}$ Step 2: Square each: $\left(-\frac{3}{2}\right)^2 = \frac{9}{4}$ $\left(\frac{8}{3}\right)^2 = \frac{64}{9}$ $\left(\frac{9}{4}\right)^2 = \frac{81}{16}$ $\left(-\frac{2}{3}\right)^2 = \frac{4}{9}$ Step 3: Multiply all: $\frac{9}{4} \times \frac{64}{9} \times \frac{81}{16} \times \frac{4}{9}$ Simplify stepwise: $\frac{9}{4} \times \frac{64}{9} = \frac{64}{4} = 16$ $16 \times \frac{81}{16} = 81$ $81 \times \frac{4}{9} = 81 \times \frac{4}{9} = 9 \times 4 = 36$ Answer: $36$ 7. Problem: Calculate $2\sqrt{7} + 3\sqrt{5} - 7\sqrt{5} - 5\sqrt{7}$ Step 1: Group like terms: $(2\sqrt{7} - 5\sqrt{7}) + (3\sqrt{5} - 7\sqrt{5})$ Step 2: Simplify coefficients: $-3\sqrt{7} - 4\sqrt{5}$ Answer: $-3\sqrt{7} - 4\sqrt{5}$ 8. Problem: Calculate a) $2 \times 3^3 - 3 \times 2^3$ b) $(10^3)^7 \times 10^{-3} : 10^{-?}$ (incomplete exponent in denominator, assuming typo, will solve a only) Step 1a: Calculate powers: $3^3 = 27$ $2^3 = 8$ Step 2a: Calculate expression: $2 \times 27 - 3 \times 8 = 54 - 24 = 30$ Answer a): $30$ 9. Problem: Simplify $4x^6 + 3x^8 - 6x^6 + x^8$ Step 1: Group like terms: $(4x^6 - 6x^6) + (3x^8 + x^8) = -2x^6 + 4x^8$ Answer: $-2x^6 + 4x^8$ 10. Problem: Calculate a) $\sqrt{\frac{1}{6}} \times \sqrt{96}$ b) $\sqrt{\frac{4}{3}} \times \sqrt{\frac{3}{5}} \times \sqrt{3 \frac{1}{5}}$ Step 1a: Multiply under one root: $\sqrt{\frac{1}{6} \times 96} = \sqrt{16} = 4$ Step 1b: Convert mixed number: $3 \frac{1}{5} = \frac{16}{5}$ Multiply under one root: $\sqrt{\frac{4}{3} \times \frac{3}{5} \times \frac{16}{5}} = \sqrt{\frac{4 \times 3 \times 16}{3 \times 5 \times 5}} = \sqrt{\frac{192}{75}}$ Simplify fraction: $\frac{192}{75} = \frac{64}{25}$ So: $\sqrt{\frac{64}{25}} = \frac{8}{5}$ Answer: a) $4$ b) $\frac{8}{5}$ 11. Problem: Calculate a) $10^{-11} \times 10^{-19}$ b) $\left(\frac{4}{3} a^5 b^6\right) \times \left(\frac{1}{4} a^8 b^7\right)$ c) $\left(-2x^6 y^7\right)^4$ Step 1a: Add exponents for same base: $10^{-11} \times 10^{-19} = 10^{-30}$ Step 1b: Multiply coefficients and add exponents: Coefficient: $\frac{4}{3} \times \frac{1}{4} = \frac{1}{3}$ $a^{5} \times a^{8} = a^{13}$ $b^{6} \times b^{7} = b^{13}$ So: $\frac{1}{3} a^{13} b^{13}$ Step 1c: Power of a product: $\left(-2\right)^4 = 16$ $x^{6 \times 4} = x^{24}$ $y^{7 \times 4} = y^{28}$ So: $16 x^{24} y^{28}$ Answer: a) $10^{-30}$ b) $\frac{1}{3} a^{13} b^{13}$ c) $16 x^{24} y^{28}$