![\"\"](\"\/..\/sd0.png\")
\nSince 1+0+2+9=12, we know 3 divides into it (because 3 divides into 12).<\/p>\n
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.<\/p>\n
Example: \u00a0Determine if each number can be square rooted (to a whole number). \u00a0If so, find its square root.<\/p>\n
a) \u00a0540
\nb) \u00a0576<\/p>\n
Before tackling the above problem, let’s dissect a number we know to be a perfect square.<\/p>\n
Example: \u00a0Confirm, by prime factorization, that the square root of 900 is 30.<\/p>\n
Solution: \u00a0We recall that a prime number is one that cannot be divided into smaller numbers, then break 900 down into primes:<\/p>\n
900=10×90=(2×5)(9×10)=(2×5)(3x3x2x5)=2x5x3x3x2x5<\/p>\n
Rearranging, we get<\/p>\n
900=2x2x3x3x5x5=(2x3x5)(2x3x5)<\/p>\n
We notice 900 can be broken into two identical groupings like so:<\/p>\n
900 = (2x3x5)(2x3x5)<\/p>\n
Therefore, the square root of 900 is 2x3x5=30<\/p>\n
We now know what to seek: \u00a0if a number is square rootable, its prime factorization can be organized into two equal groups. \u00a0The square root is simply the product of one of the groups.<\/p>\n
Back to our example:<\/p>\n
Determine the whole number square root (if it exists) of the following:<\/p>\n
a) 540
\nb) 576<\/p>\n
Solution:<\/p>\n
a) \u00a0First we break 540 into primes:<\/p>\n
540=10×54=(2×5)(6×9)=(2×5)(2x3x3x3)=2x2x3x3x3x5<\/p>\n
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.<\/p>\n
b) \u00a02 and 4 both go into 576. \u00a0Without a calculator, you either do it mentally or else use long division. \u00a0To get started, just break it in half:<\/p>\n
576=2×288=2(2×144)=2(2x12x12)=2(2x(3×4)(3×4))=2x2x3x4x3x4<\/p>\n
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:)<\/p>\n
Rearranging, we get<\/p>\n
576=2x2x3x3x4x4=(2x3x4)(2x3x4)<\/p>\n
Clearly, the prime factorization of 576 is separable into two equal groupings of 2x3x4. 2x3x4 = 24, so the square root of 576 is 24.<\/p>\n
If its prime factorization can be separated into three equal groupings, the number is a perfect cube:<\/p>\n
Example: \u00a0Confirm that 9261 is a perfect cube.<\/p>\n
Solution: \u00a0We’ll break this one down using short division. \u00a0Since 9+2+6+1=18, we know 9 divides into it:<\/p>\n
\nSince 1+0+2+9=12, we know 3 divides into it (because 3 divides into 12).<\/p>\n
![\"\"](\"\/..\/sd1.png\")
\nSince 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:<\/p>\n
![\"\"](\"\/..\/sd2.png\")
<\/p>\n
So we see that we can break down 9261 as follows:<\/p>\n
9261=9(1029)=9(3×343)=9(3x7x49)=(3×3)(3x7x7x7)=3x3x3x7x7x7<\/p>\n
Rearranging, we separate the prime factorization into three equal groupings:<\/p>\n
9261=3x3x3x7x7x7=(3×7)(3×7)(3×7)<\/p>\n
Therefore, 9261 is a perfect cube with cube root=3×7=21. \u00a0The cube root of 9261 is 21.<\/p>\n
Once again, short division was used to break down 9261. \u00a0For more explanation about that very handy technique, please check future posts:)<\/p>\n