Prime Factorization of 270:

To find the prime factorization of 270, we start by dividing it by the smallest prime number, which is 2. We continue dividing by 2 until we can no longer divide evenly. The quotient after each division is then divided by the next prime number, which is 3. Again, we continue dividing by 3 until we can no longer divide evenly.

The prime factorization of 270 can be expressed as:

270 = 2 * 3 * 3 * 3 * 5

Prime Factorization of 768:

To find the prime factorization of 768, we follow the same process. We divide it by the smallest prime number, 2, until we can no longer divide evenly. Then, we divide the quotient by the next prime number, 2, again until we can no longer divide evenly. Finally, we divide the quotient by the next prime number, 2, until we can no longer divide evenly.

The prime factorization of 768 can be expressed as:

768 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3

Expressing in Exponential Form:

To express these prime factorizations in exponential form, we rewrite each factor as a base raised to a power.

For 270:
270 = 2^1 * 3^3 * 5^1

For 768:
768 = 2^8 * 3^1

In exponential form, we represent the factors as the base raised to the power equal to the number of times it appears in the prime factorization.

Visually appealing format:

Prime Factorization of 270:
- Divided by smallest prime number, 2, until no longer divisible: 270 = 2 * 135
- Divided by next prime number, 3, until no longer divisible: 135 = 3 * 45
- Divided by next prime number, 3, until no longer divisible: 45 = 3 * 3 * 5
- Prime factorization of 270: 270 = 2 * 3 * 3 * 3 * 5

Prime Factorization of 768:
- Divided by smallest prime number, 2, until no longer divisible: 768 = 2 * 384
- Divided by next prime number, 2, until no longer divisible: 384 = 2 * 192
- Divided by next prime number, 2, until no longer divisible: 192 = 2 * 96
- Divided by next prime number, 2, until no longer divisible: 96 = 2 * 48
- Divided by next prime number, 2, until no longer divisible: 48 = 2 * 24
- Divided by next prime number, 2, until no longer divisible: 24 = 2 * 12
- Divided by next prime number, 2, until no longer divisible: 12 = 2 * 6
- Divided by next prime number, 2, until no longer divisible: 6 = 2 * 3
- Divided by next prime number, 3, until no longer divisible: 3 = 3
- Prime factorization of 768