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.

a)  540
b)  576

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:

900=10×90=(2×5)(9×10)=(2×5)(3x3x2x5)=2x5x3x3x2x5

Rearranging, we get

900=2x2x3x3x5x5=(2x3x5)(2x3x5)

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) 540
b) 576

Solution:

a)  First we break 540 into primes:

540=10×54=(2×5)(6×9)=(2×5)(2x3x3x3)=2x2x3x3x3x5

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:

576=2×288=2(2×144)=2(2x12x12)=2(2x(3×4)(3×4))=2x2x3x4x3x4

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

576=2x2x3x3x4x4=(2x3x4)(2x3x4)

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:

9261=9(1029)=9(3×343)=9(3x7x49)=(3×3)(3x7x7x7)=3x3x3x7x7x7

Rearranging, we separate the prime factorization into three equal groupings:

9261=3x3x3x7x7x7=(3×7)(3×7)(3×7)

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.