List all prime factors of: a) 144 b) 90. Simplify and classify as rational or irrational: a)  25 b)  4096

$$144 \div 2 = 72$$ $$72 \div 2 = 36$$ $$36 \div 2 = 18$$ $$18 \div 2 = 9$$

Sasa 9 haiwezi kugawanywa na 2.

$$9 \div 3 = 3$$ $$3 \div 3 = 1$$

Vigawo tasa vya 144 ni $2 \times 2 \times 2 \times 2 \times 3 \times 3$.
Hii inaweza pia kuandikwa kama $2^4 \times 3^2$.

Jibu: Vigawo tasa vya 144 ni $\boxed{2, 3}$ (au $2^4 \times 3^2$)

b) 90

$$90 \div 2 = 45$$

Sasa 45 haiwezi kugawanywa na 2.

$$45 \div 3 = 15$$ $$15 \div 3 = 5$$

Sasa 5 haiwezi kugawanywa na 3.

Vigawo tasa vya 90 ni $2 \times 3 \times 3 \times 5$.
Hii inaweza pia kuandikwa kama $2 \times 3^2 \times 5$.

Jibu: Vigawo tasa vya 90 ni $\boxed{2, 3, 5}$ (au $2 \times 3^2 \times 5$)

Swali la 19: Rahisisha na uainishe kama nambari tasa (rational) au isiyo tasa (irrational):

a) $\sqrt{2.25}$

$$\sqrt{2.25} = \sqrt{\frac{225}{100}}$$ $$\sqrt{\frac{225}{100}} = \frac{\sqrt{225}}{\sqrt{100}}$$ $$\frac{\sqrt{225}}{\sqrt{100}} = \frac{15}{10}$$

$$\frac{15}{10} = \frac{3}{2}$$

Kwa kuwa $\frac{3}{2}$ inaweza kuandikwa kama sehemu ya nambari kamili mbili (ambapo denominator si sifuri), ni nambari tasa.

Jibu: $\boxed{1.5}$ au $\boxed{\frac{3}{2}}$, na ni nambari $\boxed{\text{tasa (rational)}}$.

b) $\sqrt{0.4096}$

$$\sqrt{0.4096} = \sqrt{\frac{4096}{10000}}$$ $$\sqrt{\frac{4096}{10000}} = \frac{\sqrt{4096}}{\sqrt{10000}}$$

Tunajua $\sqrt{10000} = 100$.
Ili kupata $\sqrt{4096}$, tunaweza kutumia factorization au kukokotoa moja kwa moja.
$60^2 = 3600$ na $70^2 = 4900$. Nambari inaishia na 6, hivyo mzizi wake wa mraba unaweza kuishia na 4 au 6.
Jaribu $64^2 = 4096$.
Hivyo, $\sqrt{4096} = 64$.

$$\frac{\sqrt{4096}}{\sqrt{10000}} = \frac{64}{100}$$

$$\frac{64}{100} = \frac{16}{25}$$

Kwa kuwa $\frac{16}{25}$ inaweza kuandikwa kama sehemu ya nambari kamili mbili, ni nambari tasa.

Jibu: $\boxed{0.64}$ au $\boxed{\frac{16}{25}}$, na ni nambari $\boxed{\text{tasa (rational)}}$.