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Simplifying Radical Expressions

In this video, we are going to simplify simple radical expressions. Note that a perfect square number has a rational square root.

\sqrt{4}=2 because 2^2=4,
\sqrt{9}=3 because 3^2=9,
\sqrt{16}=4 because 4^2=16, and so on…

There are other radical expressions other than perfect square ones, to which they may not have a rational square root.
\sqrt{5} and \sqrt{20}, and \sqrt{32} are some examples.

However, having an irrational square doesn’t mean that the radical expression cannot be simplified.
\sqrt{5} cannot be simplified because it is in its simplest state.

For \sqrt{20}, 4 and 5 are factors of 20. Choosing factors that are perfect squares make the simplifying process easier.
\sqrt{20} = \sqrt{4}\times\sqrt{5}
\sqrt{4} is 2, so \sqrt{4}\times\sqrt{5} is really just 2\sqrt{5}

For \sqrt{32}, 4 and 8 are factors of 32.
\sqrt{32} = \sqrt{4}\times\sqrt{8}
\sqrt{4} is 2, so \sqrt{4}\times\sqrt{8} is 2\sqrt{8}
\sqrt{8} can be further simplified into \sqrt{4} and \sqrt{2}, or 2\times\sqrt{2}
In the end, \sqrt{32} is 4\sqrt{2}

Another way of simplifying \sqrt{32} is to use \sqrt{16} and \sqrt{2} as the factors.
Since 16 is a perfect square, \sqrt{16}\times\sqrt{2} can be written as 4\sqrt{2}, leading to the answer immediately.

Now let’s try three more examples:
\sqrt{75}
25 is a perfect square that goes into 75 and the other factor is 3
The square root of 25 is 5 so the final answer is 5\sqrt{3}

\sqrt{48}
4 and 12 go into 48
\sqrt{4} can be broken down into 2 and \sqrt{12} can be broken down into \sqrt{4} and \sqrt{3}
The square root of 4 is 22\times2\times\sqrt{3}, so we have 4\sqrt{3}

Now let’s do the same problem by using the perfect square
16 is a perfect of 48 and the accompanying factor is 3
The square root of 16 is 4, so we have 4\sqrt{3}, same answer

\sqrt{63}
9 is a perfect square that goes into 63 and the other factor is 7
The square root of 9 is 3 so the final answer is 3\sqrt{7}