Simplifying Radical Expressions 04:52 minutes

Video Transcript

Transcript Simplifying Radical Expressions

Long, long ago, somewhere in the deep blue of the Caribbean Sea. Two rival pirate captains were madly in love, and as you can imagine, this was quite a complicated situation. To keep their dalliance a secret, they pretended to be mortal enemies, and to prove this, they constantly pretended to try to overpower the other. Captain Bonny hoisted her sails due east while Calico Jack headed south. Bonny had the wind at her back, so she traveled twice as fast as Jack. Only two hours later and with a heavy heart, she wondered how far she was from her paramour’s sloop.

Simplifying radical expressions

Being a math whiz, the lovelorn captain used simplified radical expressions to figure out the distance. Let’s look at her calculations: Bonny travelled a distance equal to 4x, and Jack travelled a distance equal to 2x. Notice the right angle?

Because the points are in the shape of a right triangle, she used the Pythagorean Theorem to solve for the unknown length. Remember, the Pythagorean Theorem is: a² + b² = c² and 'c' is the hypotenuse, which is the longest side. The hypotenuse is always located opposite the right angle. Now, by subbing in the lengths she knows, Bonny can calculate the unknown length by finding the square root of 20 x².

The notation of roots

Let’s investigate the proper notation for roots. I bet you always wondered about this. The small number is called the index, and it indicates the root. A root of 2 indicates the second root, or the square root; if no number is present, the second root is assumed. The radical is the boxy shape, and the radicand is the number under the box. Understanding and using the correct terms for math can make calculations easier!

Two Properties to solve problems

There are two properties that will help us to solve problems that include square roots. First, let’s review the Product Property of Square Roots:

The square root of the product of 'ab' is equal to the square root of 'a' times the square root of 'b'.

Let’s take a look at an example: what is the square root of 27? First, factor out any perfect squares. In case you forgot, a perfect square is a number that is the square of a rational whole number. Perfect squares are 4, 9, 16, 25, and so on. Back to root 27. Using the Product Property of Square Roots, we can factor out the perfect square of 9. Then, just do the math. The end result is 3 times the square root of 3, or simply, three root three.

The second property is the Quotient Property of Square Roots. The square root of the fraction 'a/b' is equal to the square root of 'a' divided by the square root of 'b'. Let’s sub in some numbers to make this easier to understand. The square root of the fraction 49 over 81 is equal to the square root of 49 over the square root of 81 or 7 over 9.

Back to Captain Bonny. How did she simplify the radical expression of the square root of 20 x²? First, she looked for perfect square factors of 20, oh there’s one: 4! x² is obviously a perfect square, so we can group these two terms together. The rest of the calculations are easier than firing a cannon. The square root of 20x² can be simplified to 2x times the square root of 5.

And speaking of firing a cannon. Is this for real? Oh, so that was her plan all along!