What are the steps for simplifying radicals?
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How to simplify radicals
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For example 2^3=8 and ∛8 = 2 3^3= 27 and ∛27=3
The check mark √ is called the radical symbol and gives you three pieces of important information, the degree, the radical symbol, and the radicand.
The Degree is the number of times the number ( called the radicand) has been multiplied by itself.
A degree of 3 is a cube root, a 4 is the fourth root. Most times for a square root a degree will be missing, and it is assumed that it is a square root.
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Radicand The value that you are taking the root of.
Radical symbol. The symbol that looks like a check mark. √ The bar tells you how much of the value to use. You can think of the bar as parenthesis. For example,
The bar is over 30 -3 so subtract 30 -3 =27,then take the cube root. The bar is only over the 27 so take the cube root of 27 which equals 3 then subtract 2 Your radical is in the simplest form when the radicand cannot be divided evenly by a perfect square.
In order to simplify a square root you take out anything that is a perfect square.
Being familiar with the following list of perfect squares will help when simplifying radicals. Perfect Squares
4 = 2x2 9= 3x3 16= 4x4 25= 5 x5 36= 6x6 49= 7x7 64= 8x8 81= 9x9 100=10 x 10 Perfect Squares Radicals (square roots)
√4 = 2 √9 = 3 √16 = 4 √25 =5 √36 =6 √49 = 7 √64 =8 √81 =9 √100 =10
Memorizing perfect cubes will also help when simplifying radicals
8 = 2 x 2 x 2 125 = 5 x 5 x 5 512 8 x 8 x 8
27= 3 x 3 x3 216 = 6 x 6 x 6 729 = 9 x 9 x 9
64 = 4 x 4 x 4 343 = 7 x 7 x 7
Simplifying Radicals “ Square Roots”
Method 1. Find the largest perfect square that will divide into the radical
Simplify √32
Step 1 Find the largest perfect square that will divide evenly into 32 16 * 2 =32
Step 2 Write these numbers under the radical √(16*2)
Step 3 Give each number it’s own radical sign √16*√2
Step 4 “Take out” or reduce the perfect square 4√2 ( perfect square of 16=4)
Simplify √75
25*3 =75 Step 1. Find largest perfect square
Rewrite the radical as a product of the square root of 4 (found in last step) and its matching factor(2)
Simplify.
Find the largest perfect square that is a factor of the radicand (just like before)
When we simplify square roots, it is important to understand the fundamentals using the basics.
What is this expression equal to?
It is equal to 3, but why? Here is the reason.
So, we can see squaring is an inverse operation with square roots. Similarly, look at this problem with a cube root.
The answer is 4. Here is the reason.
It is evident that cubing and cube rooting are inverse operations.
Parts of a Radical
Radicals have several parts. Let us review the parts of a radical expression.
The symbol that looks like a checkmark is called a radical. A number under the radical, marked with the letter 'r,' refers to the radicand. The small number written by the checkmark, marked with the letter 'n,' refers to the index.
If an index is not written, it is assumed to be a 2 and is a square root radical.
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