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Simplifying Radicals

You already know that when a number is under a radical (e.g., $\sqrt{49}$), you're supposed to find the square root of the number (the number that, when multiplied by itself, equals the number under the radical).

You also know that most numbers are not perfect squares, so it's nearly impossible for the human mind to calcluate most square roots.

However, we can simplify many square roots -- even if we can't calculate them precisely.

For instance, if a number under a radical can be factored into perfect squares, we can find the square roots of the perfect squares, and simplify the radical. Essentially, when you have a number under a radical, you can factor that number out and square root any factor of the number.  Once you square root a factor, you remove that factor from the radical, and that number just because a multiplier of the radical (sitting just outside the radical).  The process should make more sense as you read the examples.

What does that mean?

Let's start by reviewing factoring:

Factoring is breaking a number down into its factors -- or the numbers that are multiplied together to form the number.

All numbers, except prime numbers, can be factored (a prime number can only be factored into 1 and itself, so 7 is factored to $7 \times 1$).

Example:

$12=2 \times 6$

Twelve can be factored into $2 \times 6$.

However, because 6 is not prime, $2 \times 6$ is not the "prime factorization" of 12.  

The prime factorization reduces a number to only prime factors (factors that are also prime numbers).

$$\eqalign{12&=2 \times 6\\&=2 \times 2 \times 3\\&= 2^2 \times 3}$$

The prime factorization shows that 12 contains a square number.  That's helpful to us when we are trying to simplify a radical.

Read the Prime Factorization lesson for more on factoring.

 

So, once you remember how to do a prime factorization, how do you use that to simplify a radical.  Let's use $\sqrt{12}$ as an example.

Example:

Simplify $\sqrt{12}$

$$\eqalign{\sqrt{12}&=\sqrt{2\times6} &&\text{Start by factoring}\\&=\sqrt{2\times 2 \times 3}\\&=\sqrt{2^2\times 3}&& \text{Keep factoring until you reach a prime factorization}\\&=\sqrt{2^2} \times \sqrt{3}&&\text{You can put factors under their own radicals}\\&=2 \times \sqrt{3} &&\text{If you have a perfect square, square root it and get rid of the radical}\\&=2\sqrt{3}&&\text{The radical is simplified}}$$

Essentially, the rule is, any time a number under a radical has a square root, you can find that square root and take it out from under the radical.

Let's look at another example:

Simplify $\sqrt{48}$

$$\eqalign{\sqrt{48}&=\sqrt{6\times 8} &&\text{Start by factoring}\\&=\sqrt{2\times 3 \times 2 \times 4}\\&=\sqrt{2\times 3 \times 2 \times 2 \times 2}&&\text{Keep factoring until you reach a prime factorization}\\&=\sqrt{2^2\times 2^2 \times 3}&& \text{Group factors that are the same in squares (put the 4 twos in 2 } 2^2 \text{ groups)}\\&=\sqrt{2^2} \times \sqrt{2^2} \times \sqrt{3}&&\text{You can put factors under their own radicals}\\&=2 \times 2  \times \sqrt{3} &&\text{If you have perfect squares, square root them and get rid of the radicals}\\&=4\sqrt{3}&&\text{Multiply factors that are left (inside or outside the radical)}}$$

 

One more example with a larger number:

Simplify $\sqrt{6075}$

$$\eqalign{\sqrt{6075}&=\sqrt{81\times 25 \times 3} &&\text{Start by factoring}\\&=\sqrt{9 \times 9\times 5 \times 5 \times 3}&&\text{You can go all the way to prime factorization, but because there are 2 9s, you can also leave the 9s}\\&=\sqrt{9^2\times 5^2 \times 3}&& \text{Group factors that are the same in squares}\\&=\sqrt{9^2} \times \sqrt{5^2} \times \sqrt{3}&&\text{You can put factors under their own radicals}\\&=9 \times 5 \times \sqrt{3}&&\text{If you have perfect squares, square root them and get rid of the radicals}\\&=45\sqrt{3}&&\text{Multiply factors that are left (inside or outside the radical)}}$$

To check to see if you have simplified correctly, you can square the multiplier out side the radical and multiply that by the number under the radical.  Your product should be your original radical. Take the example above:

$\eqalign{45\sqrt{3} = \sqrt{45^2 \times 3} = \sqrt{6075}}$

That radical was simplified correctly.

 

Overall, any time you want to simplify a radical, factor the number and find any perfect squares.  Square root those squares and put the roots in front of the radical. Once you have "square rooted" everything that can be done evenly, you leave the remaining factors under the radical.  Your radical is completely simplified.

Practice Problems:

  • Simplifying Radicals

    Simplify the radicals:

    1. $\sqrt{90}$
    2. $\sqrt{24}$
    3. $\sqrt{18}$
    4. $\sqrt{125}$
    5. $\sqrt{50}$
    6. $\sqrt{98}$
    7. $\sqrt{72}$
    8. $\sqrt{48}$
    9. $\sqrt{32}$
    10. $\sqrt{128}$
    11. $\sqrt{52}$
    12. $\sqrt{338}$

    Answer Key:

Skill:

Common Core Grade Level/Subject

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