Roots by Prime Factorization

A square number is two equal factors

A square number is a number multiplied by itself. So the square root of a square number is the number that, multiplied by itself, gives it: $\sqrt{144}=12$√144 = 12, because 12 × 12 = 144.

For a small square number you can remember the root. For a large one, split the number into its prime factors. The primes show the two equal factors, and one of them is the square root.

Start with 144. It is 12 × 12, and each 12 splits into 3 × 4, and each 4 into 2 × 2. So 144 = 2 × 2 × 3 × 2 × 2 × 3.

The two branches of 144 are the same: each 12 ends in the primes 2, 2 and 3.

Halve each index

Collect the primes with indices (also called exponents): there are four 2s and two 3s, so $144=2^4\times 3^2$144 = 2⁴ × 3². To find the square root, share the primes into two equal groups. Each group gets half of the 2s and half of the 3s, which is two 2s and one 3.

So $\sqrt{144}=2^2\times 3=4\times 3=12$√144 = 2² × 3 = 4 × 3 = 12. Sharing the primes into two equal groups halves every index: $2^4$2⁴ becomes $2^2$2², and $3^2$3² becomes $3^1$3¹.

The same works for 36. It splits into 2 × 2 × 3 × 3, which is $2^2\times 3^2$2² × 3². Halve each index to get 2 × 3 = 6, so $\sqrt{36}=6$√36 = 6. Check it: 6 × 6 = 36.

Drag the corner until the square holds 36 tiles. Its side is 6, which is 2 × 3, one 2 and one 3 from $2^2\times 3^2$2² × 3².

A number too big to remember

Try 784. Divide by 2 again and again: 784 ÷ 2 = 392, then 196, then 98, then 49. That is four 2s, and 49 = 7 × 7. So $784=2^4\times 7^2$784 = 2⁴ × 7².

Halve each index: $\sqrt{784}=2^2\times 7=4\times 7=28$√784 = 2² × 7 = 4 × 7 = 28. Check it: 28 × 28 = 784.

If an index is odd, the primes cannot be shared into two equal groups, so the number is not a square number. $72=2^3\times 3^2$72 = 2³ × 3², and three 2s do not split into two equal groups, so $\sqrt{72}$√72 is not a whole number.

A cube root takes a third

A cube number is a number multiplied by itself three times, so its primes can be shared into three equal groups. 1728 = 12 × 12 × 12, and each 12 is 2 × 2 × 3, so $1728=2^6\times 3^3$1728 = 2⁶ × 3³.

Share the six 2s and the three 3s into three equal groups. Each group gets two 2s and one 3, so $\sqrt[3]{1728}=2^2\times 3=12$∛1728 = 2² × 3 = 12. A cube root divides every index by 3: $2^6$2⁶ becomes $2^2$2², and $3^3$3³ becomes $3^1$3¹.

1728 splits into 12 × 144, and 144 into 12 × 12: three 12s, each ending in 2, 2 and 3.

Two common mistakes

Halving the number is not taking its square root. Half of 144 is 72, and 72 × 72 is far more than 144. It is the indices that are halved, not the number.

Taking each prime only once is also wrong. For $144=2^4\times 3^2$144 = 2⁴ × 3², the answer is not 2 × 3 = 6, because 6 × 6 = 36. Each group gets half of each index, so the 2 appears twice: $2^2\times 3=12$2² × 3 = 12.