Review Summary-Radicals
 You need to know:
 how to change entire radicals to mixed radicals (simplified form)
 how to change mixed radicals to entire radicals
 adding/subtracting radicals
 multiplying/dividing radicals
 rationalizing denominators
 solving radical equations (roots, extraneous roots)
 determining restrictions on variables
 Practice:
1) Change each entire radical to a mixed radical. (Simplified form) Look for the __________ _______ for
square roots and _____________ _________ for cube roots.
a)
50
b)
24
c)  108
d)
e)
x5
3
128
f)
16
g)  3 54
h)
f) 53 2
g)  33 6
h) x3 x 2
3
3
x5
2) Change each mixed radical back to an entire radical.
a) 2 10
b) 5 3
c)  8 2
d) x 4 x
e) 23 3
3) Add/subtract the radicals. Express the final answer in simplest form.
a)
2 5 2
e) 3 50  2 32
b)
c)
36 34 3
f) 3 20  2 12 
12 
d)
27
28  27  63  300
45
Radicals must be ____________ first before you can add/subtract them. Only _________ radicals can
be added.
g) Determine the perimeter of the shape below and express the answer in simplest form. Note: the “m”
stands for metres.
4) Multiply/Divide the radical expressions. Express final answer in simplified form.
Multiply/divide __________ and coefficients and _________ and radicands
  
a) 2 3 5 2
b)
 15  5 
 
c) 3 7
d)
 5 5 10 4
coefficient radicand
 
e) 4 7 2 14

f)
22
2
g)
10 15
2 3
h)
24
6
i)
14 12
2 3
j)
20 6
8
k) Determine the area of the rectangle (A = (l)(x)) in simplest form.
5) Rationalize each denominator.
This means there CANNOT be a _____________ in the denominator in the final answer.
To remove the radical from the denominator __________ both numerator and denominator by the
________ you want to get rid of. Sometimes it helps to __________ the radical in the denominator first
(see #c ).
a)
2
3
b)
2
2 10
c)
6 5
8
d)
2 7
3 12
6) Determining restrictions on variables in radical equations.
Note: Expression  3x  5
Equation  3x  5  7
The value of the radicand (under the radical sign) CANNOT be equal to _________ than ______
a)
x  8  14
d)
15x  6  20
so x + 8 ≥ 0 isolate “x”
b)
x  12  7
c)
3x  5  7
7) Solving Radical Equations
Radical equations can have ____ root(solution) or an ____________ root (not a solution). You must
______________ your answer back into the equation to see if the answer satisfies the equation
(______ ______ = _______ _______)
The ____________ on the variable must be stated. They can help you see if a root is extraneous.
Always isolate the radical first.
a)
x  8  6 Restrictions:
Verify:
Roots:
b)
x  3  5  2 Restrictions:
Verify:
Roots:
c)
2x  1  5 Restrictions:
Verify:
Roots:
d)
x  9   2 Restrictions:
Verify:
Roots:
Key:
1) perfect squares, perfect cubes a)
5 2
b)
2 6 c)  6 3 d) x 2 x e) 43 2 f) 23 2 g)  33 2 h) x3 x 2
75 c)  128 d) x 9 e) 3 24 f) 3 250 g)  3 162 h) 3 x 5
3a) 6 2 b)  3 c) 5 3 d) 5 7  7 3 e) 7 2 f) 9 5  4 3 g) 10 2  4 3
2a)
40 b)
4) coefficient, radicands a)
10 6 b) 5 3 c) 3 7 d) 100 2 e) 56 2 f) 11 g) 5 5 h) 2 i) 14 j) 10 3 k) 30 2
6 b) 5 c) 3 10 d) 7
3
10
2
9
2
5
6) less than zero, a) x ≥ -8 b) x ≥ -5 c) x ≥ 12 d) x ≥ 
e) x ≥
3
5
5) radical, multiply, radical, simplify a)
7) one, extraneous, substitute, left side, right side a) restriction x ≥ -8 root: x = 28 b) restriction x ≥ 3 root: x = 45
c) restriction x ≥ -5 root: x = 12 d) restriction x ≥ 9 extraneous root: x = 13