Zawadi travelled 512 km in 8 hours to visit relatives. Express distance and time as products of prime factors and use division law of indices to find speed. Leave answer in index form. (5 marks)

Tunatafuta vigawo tasa vya 512.

$$512 = 2 \times 256$$ $$256 = 2 \times 128$$ $$128 = 2 \times 64$$ $$64 = 2 \times 32$$ $$32 = 2 \times 16$$ $$16 = 2 \times 8$$ $$8 = 2 \times 4$$ $$4 = 2 \times 2$$

Hivyo, $512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9$.

Tunatafuta vigawo tasa vya 8.

$$8 = 2 \times 4 = 2 \times 2 \times 2 = 2^3$$

Kasi hupatikana kwa kugawanya umbali kwa muda.

$$\text{Kasi} = \frac{\text{Umbali}}{\text{Muda}}$$ $$\text{Kasi} = \frac{512 \text{ km}}{8 \text{ hours}}$$

Badilisha umbali na muda kwa vigawo vyao tasa:

$$\text{Kasi} = \frac{2^9}{2^3} \text{ km/h}$$

Tumia kanuni ya kugawanya ya vipeo, ambayo inasema $a^m \div a^n = a^{m-n}$:

$$\text{Kasi} = 2^{9-3} \text{ km/h}$$ $$\text{Kasi} = 2^6 \text{ km/h}$$

Jibu: Kasi ni $2^6 \text{ km/h}$