Simplifying, Adding and Multiplying Radicals

0:00A radical is a root sign over a number called the radicand. How do we simplify the square root of 72?

0:08Look for perfect squares, whole numbers times themselves, like 4, 9, 16, 25 and 36.

0:17The largest one that is a factor of 72 is 36, since 72 is 36 times 2.

0:25The root of a product is the product of the roots, so root 72 is root 36 times root 2.

0:34Root 36 is 6, so root 72 equals 6 root 2.

0:40Check: 6 times 6 is 36, and 36 times 2 is 72.

0:47Now add 2 root 12 and root 27. The radicands differ, so simplify each one first.

0:5512 is 4 times 3, so root 12 is 2 root 3, and 2 root 12 is 2 times 2 root 3.

1:0427 is 9 times 3, so root 27 is 3 root 3.

1:102 times 2 root 3 is 4 root 3.

1:14Both now have radicand 3, so they are like radicals. Just as 4x plus 3x is 7x, they add to 7 root 3.

1:26Multiply the outside numbers, called coefficients, together, and the inside numbers, the radicands, together.

1:342 times 4 is 8, and 3 times 6 is 18, so 2 root 3 times 4 root 6 is 8 root 18.

1:4418 is 9 times 2, so root 18 is 3 root 2, and 8 times 3 root 2 is 24 root 2.

1:56Careful: root 2 plus root 3 is not root 5, because they are not like radicals.

2:03Root 2 plus root 3 is about 3.15, but root 5 is only about 2.24.

2:12To recap: pull out perfect squares, add only like radicals, and multiply outsides together and insides together.