4.1
Mixed and Entire Radicals
Math 20-2
A radical is simply the ROOT SIGN. √ . The root sign is just a symbol, like + or ÷,
which holds a specific meaning in math. There are different types of root signs:
√
𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡
3
𝑐𝑢𝑏𝑒𝑑 𝑟𝑜𝑜𝑡
√
4
√ 𝑓𝑜𝑢𝑟𝑡ℎ 𝑟𝑜𝑜𝑡 … etc.
Square roots
Square roots are the values that when multiplied to themselves results in the radicand.
±√25 = ±5
Square root
Perfect square
(Radicand)
Remember that when we are dealing with square roots, there are 2 values that
fulfill the definition of “a number multiplied by itself”
√25 = 5 and -5
Principle Root
Secondary Root
How the secondary root works: unless specifically asked for in the question, you
will ALWAYS use the principle root. The difference in how the questions will
look is this:
√25 = 5
or
−√25 = −5
The only time you will have to consider both the positive and the negative roots is
if you are doing a word problem and you are required to take the square root of
both sides during your solution:
3x2 = 27
Entire radical – all values are under the root sign.
√150
√27
Mixed radical – combination of a whole number coefficient and a root
5√6
3 √3
Simplifying a radical – converting an entire radical to a mixed radical, if possible.
 To do this you must look to see if one of the factors of the number is a
perfect square.
 If it is, actually convert it to a whole number and leave the remaining factor
under the radical.
√150
√27
4.2
Adding and Subtracting Radicals
Math 20-2
Adding and subtracting radicals is very similar to adding and subtracting variables. You
can only collect like terms. This means you must simplify all entire radicals into mixed
radicals, and then collect like radicals.
4.3
Multiplying and Dividing Radicals
Math 20-2
Multiply
1. Single Term:
(√5)(√3)
=√15
2. Distribution
3. Distribution (mixed)
4. Foil
***always ask yourself if you can simplify
√3(√7 + √6)
4√6(7√5 + √3)
(√2 + √3)(√5 − √6)
Divide
It is improper form to have a radical in the denominator of the fraction. This means that
if needed, you will have to RATIONALIZE the denominator. This means mathematically
remove the radical.
6√40
2√5
12√15
5√12
4.4
Simplifying Radicals
Math 20-2
Restrictions – values that do not allow the answer to be defined
√𝑥
x≥0 and x€R
*cannot take the square root of a negative number
√𝑥 2
x€R
*by squaring the root, you remove any restrictions
3
x€R
*you can take the cubed root of any number.
x>0 and x€R
*you cannot divide by 0
√𝑥
1
√
𝑥
**if the root is EVEN then there is a restriction stating that the root must be greater than
or equal to 0
**if the root is ODD then there is no restriction.
**if the root is complex, determine what value “x” cannot be.
Absolute Value – is a way to state a value based on its distance from 0 on the number
line.
lxl = x and l-xl = x.
4.5/4.6
Exploring Radical Equations
When dealing with equations, you must remember to isolate the variable.





Opposite math
What you do to one side you must do to the other
Leave the radical to the end
State restrictions
Verify each side independently
√3𝑥 = 6
Verify:
√𝑥 − 1 + 3 = 4
Verify:
√𝑥 + 2 = −3
Verify:
Math 20-2
5√𝑥 + 3 = 15
Verify:
The forward and backward motion of a swing can be modelled using the formula:
𝐿
𝑇 = 2𝜋√9.8
Where t is the time in seconds and L is the length (m) of chain supporting the
swing. What length of chain is needed for the swing if it takes 2.5 s for the swing
to become still?