The math tutor continues to appreciate prime factorization for all it yields.
Let’s imagine you need to determine the square root of a number without a calculator. This challenge is part of the curriculum for local high school students.
Example: Determine if each number can be square rooted (to a whole number). If so, find its square root.
Before tackling the above problem, let’s dissect a number we know to be a perfect square.
Example: Confirm, by prime factorization, that the square root of 900 is 30.
Solution: We recall that a prime number is one that cannot be divided into smaller numbers, then break 900 down into primes:
Rearranging, we get
We notice 900 can be broken into two identical groupings like so:
900 = (2x3x5)(2x3x5)
Therefore, the square root of 900 is 2x3x5=30
We now know what to seek: if a number is square rootable, its prime factorization can be organized into two equal groups. The square root is simply the product of one of the groups.
Back to our example:
Determine the whole number square root (if it exists) of the following:
a) First we break 540 into primes:
With only one 5 in the prime factorization, we can’t separate it into two equal groups. 540 doesn’t have a whole number square root.
b) 2 and 4 both go into 576. Without a calculator, you either do it mentally or else use long division. To get started, just break it in half:
Since we have only multiplication here, we can add and rearrange brackets at will. However, with mixed operations we wouldn’t be able to do so:)
Rearranging, we get
Clearly, the prime factorization of 576 is separable into two equal groupings of 2x3x4. 2x3x4 = 24, so the square root of 576 is 24.
If its prime factorization can be separated into three equal groupings, the number is a perfect cube:
Example: Confirm that 9261 is a perfect cube.
Solution: We’ll break this one down using short division. Since 9+2+6+1=18, we know 9 divides into it:
Since 1+0+2+9=12, we know 3 divides into it (because 3 divides into 12).
Since 3+4+3=10 (which 3 doesn’t divide into), and 343 doesn’t end in 5 or 0, the next number to try is 7:
So we see that we can break down 9261 as follows:
Rearranging, we separate the prime factorization into three equal groupings:
Therefore, 9261 is a perfect cube with cube root=3×7=21. The cube root of 9261 is 21.
Once again, short division was used to break down 9261. For more explanation about that very handy technique, please check future posts:)
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.