3-8 Finding & Estimating Square Roots 11-1 Simplifying Radicals

3-8 Finding & Estimating Square Roots11-1 Simplifying Radicals

Vocab • Radical – symbol used to indicate a square root • Radicand– the expression under the radical • Perfect square – a number whose square root is an integer • To simplify means to find another expression with the same value. It does not just mean to find a decimal approximation

Finding the square root of a number is the equivalent of “unsquaring” it. This is a term I made up – but it makes sense. Start with a length 3 ft Now square it – or make a square out of it! It’s area is 9 ft2 – when you square 3 you get 9.

To unsquare a number, or find its square root, start with a square… 25 ft2 Now unsquare it – or figure out how long each side is. Each side is 5 ft, the square root of 25 is 5.

Some squares are perfect…this means their square root (or side length) is an integer. 1 4 9 16 25 36 49 64 81 100 The first ten perfect squares are:

Why are these perfect? 1 4 9 16 25 36 49 64 81 100 = = = = = = = = = = 12 22 32 42 52 62 72 82 92 102 OR square unsquare

You can estimate square roots by determining which “perfect square” they are closer to…. Of course you can also use a calculator! Is between & Since 11 is closer to 9 than it is to 16, I would estimate is about 3

Simplifying radicals….this is in Chapter 11, but we won’t get to Chapter 11 and this is a great place to talk about it! To simplify means to find another expression with the same value. It does not just mean to find a decimal approximation…especially with radicals!

Let’s look at some radical properties! This might look easier with numbers… We can prove this one… 6 = 3 * 2

Here’s another radical property! This might look easier with numbers too We can prove this one using a calculator.

In order to simplify radicals, you look at the number under the radical aka the radicand, which isn’t perfect, and try to find perfection! 45 is NOT perfect, but it has a factor that IS perfect…9 Rewrite 45 as 9 * 5

The perfect squares aren’t always so easy to see…it can be helpful to make a list of them to refer to and use a calculator to check for perfect factors! Simplify :

It works the same way with variables! x2 is a perfect square…its square root is x x3 is NOT perfect, but it has a factor that IS perfect… x2 Rewrite x3 as x2 * x

Let’s try combining numbers and letters! Simplify : 4) 1) 5) 2) 6) 3)

4) 90a3bc2 1) 12x2 5) 2) 6) 3)

3-8 Finding & Estimating Square Roots 11-1 Simplifying Radicals