Let's break down each problem from the image.

E2. Express each of the following as a product of prime factors only in exponential form:

i) 108 × 192

We'll factorize each number:

• $108 = 2^2 \times 3^3$
• $192 = 2^6 \times 3$

Now multiply the factorizations: $108 \times 192 = (2^2 \times 3^3) \times (2^6 \times 3) = 2^{2+6} \times 3^{3+1} = 2^8 \times 3^4$

ii) 270

Factorize 270: $270 = 2 \times 3^3 \times 5$

iii) 729 × 64

Factorize each number:

• $729 = 3^6$
• $64 = 2^6$

Now multiply the factorizations: $729 \times 64 = 3^6 \times 2^6$

E3. Find the unit digit of:

a) $3^4$

$3^4 = 81 \quad \text{(unit digit is 1)}$

b) $3^{2025}$

The powers of 3 repeat in cycles of 4 for the unit digit: $3, 9, 7, 1$. To find the unit digit of $3^{2025}$, divide 2025 by 4 and find the remainder: $2025 \div 4 = 506 \, \text{remainder} \, 1$ So, the unit digit is the same as $3^1$, which is 3.

c) $2^4$

$2^4 = 16 \quad \text{(unit digit is 6)}$

d) $2^{2024}$

The powers of 2 repeat in cycles of 4 for the unit digit: $2, 4, 8, 6$. To find the unit digit of $2^{2024}$, divide 2024 by 4 and find the remainder: $2024 \div 4 = 506 \, \text{remainder} \, 0$ So, the unit digit is the same as $2^4$, which is 6.

E4. Find the remainder for:

a) $133^2$ divided by 12

First, calculate $133 \mod 12$: $133 \div 12 = 11 \quad \text{(remainder is 1)}$ Now, square the remainder: $133^2 \mod 12 = 1^2 \mod 12 = 1$

b) $133^{2024}$ divided by 12

Since $133 \mod 12 = 1$, we have: $133^{2024} \mod 12 = 1^{2024} \mod 12 = 1$

Let me know if you'd like further details, and feel free to ask any questions!

Here are 5 related questions for practice:

1. What is the prime factorization of 540 in exponential form?
2. Find the unit digit of $4^{2025}$.
3. Factorize $360$ into prime factors in exponential form.
4. Find the remainder when $144^2$ is divided by 9.
5. Determine the remainder when $5^{50}$ is divided by 12.

Tip: For powers involving units digits, always check for cyclic patterns!