What is the smallest perfect square that has a factor of 800. What is the smallest perfect cube that has a factor of 800?
Smallest perfect square would be 1600 and smallest perfect cube I think would be 8000.
800= 2,2,2,2,2,5,5
2^2 × 2^2 × 5^2 × 2^2 (need another 2 to make it a square)
1600 perfect square
800= 2,2,2,2,2,5,5
2^3 × 2^3 × 5^3
8000 perfect cube
To find the smallest perfect square that has a factor of 800, you need to consider the prime factorization of 800, which is 2^5 * 5^2. A perfect square is a number that can be expressed as the square of an integer. So, you want to find the smallest number that has at least four 2s and two 5s in its prime factorization.
To do this, you can start by finding the prime factorization of a perfect square using the prime factors of 800. The prime factors of 800 can be arranged in pairs to make up the prime factors of a perfect square.
In this case, you have four 2s and two 5s. You can pair two 2s and one 5 to make up the prime factors of a perfect square. Therefore, you would have 2^2 * 5^2, which simplifies to 4 * 25 = 100. So, the smallest perfect square that has a factor of 800 is 100.
To find the smallest perfect cube that has a factor of 800, you need to consider the prime factorization of 800, which is 2^5 * 5^2. A perfect cube is a number that can be expressed as the cube of an integer. So, you want to find the smallest number that has at least three 2s and one 5 in its prime factorization.
To do this, you can start by finding the prime factorization of a perfect cube using the prime factors of 800. The prime factors of 800 can be arranged in groups of three to make up the prime factors of a perfect cube.
In this case, you have three 2s and one 5. You can group two 2s and one 5 to make up the prime factors of a perfect cube. Therefore, you would have 2^3 * 5^1, which simplifies to 8 * 5 = 40. So, the smallest perfect cube that has a factor of 800 is 40.
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