Pythagorean triples are nice, but most right triangles have at least one side that is not a whole number. When this happens, the length of at least one side is often an irrational number, or in other words a decimal that never ends and never repeats. This can make it difficult to get the exact length of the side of a triangle because if we left the answer in decimal form, it would have to be rounded off and would no longer exactly right. In order to avoid rounding, we must be able to leave answers in simplest radical form.

To simplify a radical, find the largest factor of the number under the radical that is a perfect square. Then, by taking the square root of that factor you will remove the factor from the radical. When the number inside the radical has no more factors that are perfect squares, the answer is in simplest radical form.

**Key Concepts: Reducing a Radical**
Determine perfect square factors of the number under the radical symbol (e.g. 4, 9, 16, 25, 36, etc.). Rewrite the radicand as the product of this perfect square number and the other factor. Find the square root of the perfect square, cross it off, and write it in front of the radical. Repeat if possible.

Adding and subtracting with radicals works just like adding and subtracting with variables. To add or subtract radicals, just combine like terms.

$3 \sqrt{5} + 4 \sqrt{5} = 7 \sqrt{5}$

When multiplying radicals, you must multiply numbers with numbers and radicals with radicals, just like you would with variables.

$2\sqrt{3}\times4\sqrt{5}=8\sqrt{15}$

$\sqrt{6} \times \sqrt{6} = \sqrt{36} = 6$

Dividing works the same way.

$\frac{4 \sqrt{6}}{2 \sqrt{2}} = 2 \sqrt{3}$

Another rule when simplifying radicals is that you can never leave a radical in the denominator of a fraction. Fortunately, it is easy to change the appearance of a fraction without changing its value. Since we do not want to change the value of a correct answer, multiply the answer by the identity which is 1. Because we want to change the appearance of the answer, choose a way to write 1 as a fraction. For example $\frac{3}{3}$ is another way to write 1. To remove a radical from the denominator of a fraction, write 1 using the radical that you want to remove.

**Key Concepts:**
Adding, subtracting, multiplying, and dividing expressions with radicals work just like you learned in algebra. Treat the radical part as if it was a variable like "x". Use your calculator to check your work– type in your original expression and get the decimal equivalent, then do the same for your simplified answer. Are they the same?

In your journal, simplify each radical expression.

$\sqrt{32}$ 2. $4 \sqrt{3} + 3 \sqrt{5} - 2 \sqrt{3} + 6 \sqrt{5}$

$4\sqrt{6}\times3\sqrt{3}$ 4. $6\times3\sqrt{3}$

5. $\sqrt{7^{2}}$ 6. $\frac{6 \sqrt{10}}{2 \sqrt{5}}$

7. $\frac{12}{\sqrt{3}}$ 8. $\frac{4 \sqrt{3}}{\sqrt{2}}$

After you have finished, click here to check your answers.