Download Lesson Notes 1-2 Simplifying/Adding/Subtracting Radicals Radicals

Lesson Notes 1-2
Simplifying/Adding/Subtracting Radicals
Radicals are the name given to finding a root of any degree. You should be familiar with
the terms square root and cube roots, but there are many other types of roots as well. We
write radicals in the form:
an x
where a is the coefficient, n is the index which tells us what root we are
looking for, and x is the radicand, which is the number being “rooted.
A mixed radical is a radical that has a number outside the square root sign.
For example, 3 5 is a mixed radical.
An entire radical is a radical that has no number outside the square root sign.
For example, 20 is an entire radical.
Every mixed radical can be expressed as an entire radical. The steps to write a mixed
radical as an entire radical are:
1. Determine the value of the coefficient to the power of the index.
2. Multiply this number by the radicand.
Example 1: Write the following mixed radicals as entire radicals.
3
2
3
(b) 5b 3b
(c) 2 3
a) 3 5
To simplify radicals means to write entire radicals as mixed radicals. The steps to write
an entire radical as a mixed radical are:
1. Break a number up into two numbers where one is a perfect square.
2. Square root the perfect square (this is the coefficient of the radical) leaving all
other numbers as the radicand.
Note 1: If there is already a coefficient multiply the new and old coefficients
Note 2: If the index is a three look for a perfect cube
Example 2: Write the following entire radicals as mixed radicals.
2
(b) 45
(c) 2 32x
a) 54
Adding and subtracting radicals is similar to adding variables:
Variables
x + x = 2x
Radicals
3+ 3 =2 3
With radicals you must have a common radical where in variables you needed a common
variable.
Example 3: Add or subtract the following.
a) 3 + 2 3
(b)
3 6+5 6
Sometimes you have to simplify each radical first and then add or subtract the radicals.
Example 4: Add or subtract the following.
a) 32 + 8
(b) 28 − 63 + 112
c) 5 28 − 3 63 + 2 112
(d)
4c − 4 9c