Write the prime factorization of 363 using exponents.
The correct answer and explanation is:
Correct Answer:
The prime factorization of 363 using exponents is:
3 × 11 × 11 = 3 × 11²
Explanation:
Prime factorization is the process of expressing a number as a product of its prime numbers. A prime number is a number greater than one that has only two factors, one and itself. Examples of prime numbers include 2, 3, 5, 7, 11, 13, and so on.
To factor 363, the first step is to test for divisibility by the smallest prime numbers, starting from 2. Since 363 is an odd number, it is not divisible by 2. The next prime number is 3. To check for divisibility by 3, add the digits of 363:
3 + 6 + 3 = 12.
Since 12 is divisible by 3, it means 363 is divisible by 3.
Now divide 363 by 3:
363 ÷ 3 = 121.
Now factor 121. Start testing again with small prime numbers. It is not divisible by 2, 3, 5, or 7. But it is divisible by 11 because 11 × 11 = 121. So:
121 = 11 × 11 = 11².
Now rewrite the full factorization:
363 = 3 × 121
= 3 × 11 × 11
= 3 × 11²
This is the prime factorization of 363 using exponents. Writing repeated prime numbers using exponents makes the expression shorter and clearer. In this case, 11 appears twice, so it is written as 11 raised to the power of 2, or 11².
Understanding prime factorization helps in many areas of mathematics such as finding the highest common factor (HCF), lowest common multiple (LCM), simplifying fractions, and solving problems involving divisibility or number properties.