Prime Factorization of 729×1024

To express the product 729×1024 as a product of primes in exponential form, we need to find the prime factors of both numbers and then combine them.

Prime Factorization of 729

To find the prime factors of 729, we can start by dividing it by the smallest prime number, which is 2. However, 729 is not divisible by 2. Next, we try dividing it by 3, which is another prime number. 729 divided by 3 gives us 243.

Now, we can continue dividing 243 by prime numbers until we reach the smallest prime number, which is 2. 243 divided by 3 gives us 81, and 81 divided by 3 gives us 27. Finally, 27 divided by 3 gives us 9, and 9 divided by 3 gives us 3.

Since 3 is a prime number, we have found all the prime factors of 729. Therefore, the prime factorization of 729 is 3×3×3×3×3, which can also be written as 3^6.

Prime Factorization of 1024

To find the prime factors of 1024, we can start by dividing it by 2, as 2 is the smallest prime number. 1024 divided by 2 gives us 512.

Next, we continue dividing 512 by 2 until we reach the smallest prime number. 512 divided by 2 gives us 256, 256 divided by 2 gives us 128, and 128 divided by 2 gives us 64. Continuing this process, 64 divided by 2 gives us 32, 32 divided by 2 gives us 16, 16 divided by 2 gives us 8, and finally, 8 divided by 2 gives us 4.

Since 4 is not divisible by 2, we stop here. Therefore, the prime factorization of 1024 is 2×2×2×2×2×2×2×2×2×2, which can also be written as 2^10.

Prime Factorization of the Product

To find the prime factorization of the product 729×1024, we can combine the prime factorization of 729 and 1024.

The prime factorization of 729 is 3^6, and the prime factorization of 1024 is 2^10. Therefore, the prime factorization of 729×1024 is (3^6)×(2^10).

Final Answer

In exponential form, the product 729×1024 can be expressed as 3^6 × 2^10.