Using an important property of radicals allows us to simplify radicals as much as possible.  Let’s state the property below.

Definition:   If a and b are any numbers, then the following property holds:

$$\sqrt a \cdot \sqrt b = \sqrt {ab}$$

We can use this property and prime factorization to simplify complicated looking radicals.  Let’s try an example.

Simplify $$10\sqrt {343{x^2}}$$.

Let’s look at the prime factorization of the number under the radical, 343.  We want to see if there are any perfect squares embedded in 343.  We have

$$343 = 7*49$$

$$= 7*{7^2}$$

There is a perfect square, 49, embedded within 343.  Now we will rewrite 343 in a convenient way to simplify.  We have

$$10\sqrt {343{x^2}} = 10\sqrt {7*{7^2}*{x^2}}$$

Now implement the property of radicals that we discussed above and simplify each individual radical one at a time.

$$10\sqrt {343{x^2}} = 10\sqrt {7*{7^2}*{x^2}}$$

$$= 10*\sqrt 7 *\sqrt {{7^2}} *\sqrt {{x^2}}$$

$$= 10*\sqrt 7 *7*\left| x \right|,$$, don't forget that $$\sqrt {{x^2}} = |x|$$

So we have simplified a radical. $$10\sqrt {343{x^2}} = 70\left| x \right|\sqrt 7$$. Why don’t we do another example?

Simplify $$2\sqrt {216{n^5}}$$. Again we want to break down 216 into its prime factorizations to see if there are any perfect squares lurking inside.  We have

$$216 = 2*108$$

$$= 2*2*54$$

$$= {2^2}*54$$

$$= {2^2}*9*6$$

$$= {2^2}*{3^2}*6$$

In fact, there are two perfect squares embedded within 216.  Let’s rewrite our radical expression and finish simplifying:

$$2\sqrt {216{n^5}} = 2\sqrt {{2^2}*{3^2}*6*{n^2}*{n^2}*n}$$

$$= 2*\sqrt {{2^2}} *\sqrt {{3^2}} *\sqrt 6 *\sqrt {{n^2}} *\sqrt {{n^2}} *\sqrt n$$, from the property of radicals

$$= 2*2*3*\sqrt 6 *\left| n \right|*\left| n \right|*\sqrt n$$

$$= 12*\sqrt 6 *\left| {{n^2}} \right|*\sqrt n$$

$$= 12{n^2}\sqrt {6n}$$, disregard the absolute value bars since $${n^2}$$ is always positive

So $$2\sqrt {216{n^5}} = 12{n^2}\sqrt {6n}$$.

As you practice more with simplifying radicals, the procedure becomes so fast that you’ll be doing most of it in your head without writing the intermediate simplifying steps.  Good luck!

2212 x

Simplify.

This free worksheet contains 10 assignments each with 24 questions with answers.

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1805 x

Simplify.

This free worksheet contains 10 assignments each with 24 questions with answers.

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1429 x

Simplify. Use absolute value signs when necessary.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

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