Write 224 as a product of powers of its prime factors.
The correct answer and explanation is:
To express 224 as a product of powers of its prime factors, we start by finding the prime factorization of 224.
- Divide by 2 (smallest prime number):
224 is an even number, so it can be divided by 2:
224÷2=112224 ÷ 2 = 112. - Divide by 2 again:
112 is also even, so we divide by 2 again:
112÷2=56112 ÷ 2 = 56. - Divide by 2 once more:
56 is still even, so divide by 2:
56÷2=2856 ÷ 2 = 28. - Divide by 2 again:
28 is even, so divide by 2:
28÷2=1428 ÷ 2 = 14. - Divide by 2 one more time:
14 is even, so divide by 2:
14÷2=714 ÷ 2 = 7. - 7 is a prime number:
At this point, 7 cannot be divided by 2, and it’s already a prime number.
Now, we can write the prime factorization of 224: 224=2×2×2×2×2×7=25×7224 = 2 × 2 × 2 × 2 × 2 × 7 = 2^5 × 7
Thus, the prime factorization of 224 is 25×72^5 × 7.
In conclusion, the product of the powers of prime factors for 224 is 25×72^5 × 7. This means that 224 can be broken down into five factors of 2 and one factor of 7, which are its prime factors raised to appropriate powers.