Powers and Prime Factorization
When working with powers of numbers, there are patterns we can use to make it easier to find prime factorizations, factors, and multiples.
Perfect Squares and Perfect Cubes
A perfect square is an integer that is a square of another integer. A perfect cube is an integer that is a cube of another integer.
The number 36 is a perfect square because 36 = 6 × 6 = 62.
The number 125 is a perfect cube because 125 = 5 × 5 × 5 = 53.
There are patterns we can use to identify perfect squares and perfect cubes simply by looking at their prime factorizations.
81 = 9 × 9 = (3 × 3) × (3 × 3) = 34
100 = 10 × 10 = (2 × 5) × (2 × 5) = 22 × 52
144 = 12 × 12 = (2 × 2 × 3) × (2 × 2 × 3) = 24 × 32
The numbers 81, 100, and 144 are all perfect squares, and the exponents in their prime factorizations are all even numbers.
That is not a coincidence. When a number’s prime factorization has exponents that are all even, the number can always be written as the square of another integer.
For example, the prime factorization of 291,600 is 24 × 36 × 52. Because the exponents in the prime factorization are all even, we can divide the prime factors of 291,600 into two equal groups:
291,600 = 24 × 36 × 52 = (22 × 33 × 51) × (22 × 33 × 51) = 540 × 540 = 5402
Conversely, any number with one or more odd exponents in its prime factorization cannot be a perfect square because it would be impossible to divide its prime factors into two equal groups.
Practice Identifying Perfect Squares and Cubes
Is a perfect square?
If we apply the same logic, we can identify a similar pattern for perfect cubes.
The prime factorization of 592,704 is 26 × 33 × 73. Because the exponents in the prime factorization are all divisible by three, we can divide the prime factors of 592,704 into three equal groups:
592,704 = 26 × 33 × 73 = (22 × 31 × 71) × (22 × 31 × 71) × (22 × 31 × 71) = 84 × 84 × 84 = 843
Is a perfect cube?
If the exponents in a number’s prime factorization are all even and divisible by two, then the number is a perfect square. If the exponents are all divisible by three, then the number is a perfect cube.
Finding Factors and Multiples (Perfect Squares and Cubes)
What is the greatest factor of 12,096 that is also a perfect square?
The prime factorization of 12,096 is 26 × 33 × 71. Any factor of 12,096 that is also a perfect square has to have a prime factorization that fits inside of the prime factorization of 12,096 with exponents that are all divisible by two. This means it can have either zero, two, four, or six 2’s in its prime factorization; zero or two 3’s; and zero 7’s. It can’t have three 3’s or one 7 because those exponents aren’t divisible by two and we wouldn’t be able to divide those prime factors into two equal groups.
The greatest factor of 12,096 that is also a perfect square is 576 (26 × 32 × 70) = 242.
What is the smallest multiple of 12,096 that is also a perfect cube?
Any multiple of 12,096 that is also a perfect cube has to have a prime factorization that can fit the prime factorization of 12,096 inside of it with exponents that are all divisible by three. This means it can have six, nine, twelve, or more 2’s in its prime factorization; three, six, nine, or more 3’s; and three, six, nine, or more 7’s. Even though a prime factorization would only need one 7 to fit the prime factorization of 12,096 inside of it, our prime factorization has to have at least three 7’s in it because we need to be able to divide the prime factors into three equal groups.
The smallest multiple of 12,096 that is also a perfect cube is 592,704 (26 × 33 × 73) = 843.