How do you calculate the square root and cube root of a number?

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The easiest way to find roots is through prime factorization. In order for a whole number to be a perfect square, each of its prime factors must have an even exponent. In other words, all of its distinct prime factors must occur in multiples of 2.

For example, `100 =...

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The easiest way to find roots is through prime factorization. In order for a whole number to be a perfect square, each of its prime factors must have an even exponent. In other words, all of its distinct prime factors must occur in multiples of 2.

For example, `100 = 2^2*5^2` and `144=3^2*2^4`

This also works backwards: if you multiply a bunch of prime numbers raised to even powers, you get a perfect square.

`3^2*5^2*7^2=11025` ; `\sqrt[11025]=105`

To find the square root of a number using prime factorization, simply divide the exponent for each prime factor by two and then multiply what's left. We can see this in the previous three examples.

`2*5=10`

`3*2^2=12`

`3*5*7=105`

Finding a cube root follows much the same reasoning. In order to be a perfect cube, each of the prime factors of an integer must be raised to a power of 3.

`216=3^3*2^3`

To find a cube root, simply divide the exponent for each prime factor by 3.

`3*2=6` and `^3\sqrt[216]=6`

In cases where the number is not a perfect square (or cube), take the root of the prime factors that are raised to an even power (or multiple of 3) and leave the rest under the root sign.

`\sqrt[12]=\sqrt[2^2]*\sqrt[3]=2\sqrt[3]`

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