Evaluate without using a calculator or table: sqrt(45 x (2.04)2) / (2.89 x sqrt(0.05)).

Tunayo $\sqrt{45 \times (2.04)^2}$.
Tunajua kwamba $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ na $\sqrt{x^2} = x$.

$$\sqrt{45 \times (2.04)^2} = \sqrt{45} \times \sqrt{(2.04)^2}$$ $$= \sqrt{9 \times 5} \times 2.04$$ $$= \sqrt{9} \times \sqrt{5} \times 2.04$$ $$= 3\sqrt{5} \times 2.04$$

Tunayo $2.89 \times \sqrt{0.05}$.
Tunajua kwamba $0.05 = \frac{5}{100}$.

$$2.89 \times \sqrt{0.05} = 2.89 \times \sqrt{\frac{5}{100}}$$ $$= 2.89 \times \frac{\sqrt{5}}{\sqrt{100}}$$ $$= 2.89 \times \frac{\sqrt{5}}{10}$$

$$\frac{3\sqrt{5} \times 2.04}{2.89 \times \frac{\sqrt{5}}{10}}$$

Futa $\sqrt{5}$ kutoka juu na chini:

$$\frac{3 \times 2.04}{2.89 \times \frac{1}{10}}$$

Tunajua $2.04 = \frac{204}{100}$ na $2.89 = \frac{289}{100}$.

$$\frac{3 \times \frac{204}{100}}{\frac{289}{100} \times \frac{1}{10}}$$ $$\frac{\frac{3 \times 204}{100}}{\frac{289}{1000}}$$

Geuza mgawanyiko kuwa kuzidisha kwa kugeuza sehemu ya chini:

$$\frac{3 \times 204}{100} \times \frac{1000}{289}$$

Futa 100:

$$\frac{3 \times 204 \times 10}{289}$$

Tunajua $289 = 17^2$ na $204 = 17 \times 12$.

$$\frac{3 \times (17 \times 12) \times 10}{17 \times 17}$$

Futa 17 moja kutoka juu na chini:

$$\frac{3 \times 12 \times 10}{17}$$ $$\frac{36 \times 10}{17}$$ $$\frac{360}{17}$$

Kama nambari mchanganyiko, hii ni $21 \frac{3}{17}$.