Radicals:
Definition: A number r is a square root of another number a if
r2 = a.
Example: 3 is a square root of 9 since 32 = 9
Example: −3 is also a square root of 9, since (−3)2 = 9
Notice that each positive number a has two square roots, one is positive
and the other is negative.
Notice that only positive numbers have real numbers as square roots.
−16, for example, does not have any real number square roots since
the square of a real number can never be negative.
Definition:
The principal square root of a number a, represented
√
by a (read radical of a ), is the non-negative square root of a.
√
Example:
The
principal
square
root
of
25
is
represented
by
25.
√
√
25 is the positive square root of 25. In other words, 25 = 5.
√
Example: 64 = 8
√
Example: 1 = 1
√
Example: 0 = 0
Note: Some people are confused about the definition of the square
root(s) of a number and the principal square root of a number. By
definition, a positive number always has two square roots, one positive
and one negative. The positive of that sqaure root is the principal
square root, and is what is represented by the radical. In other words,
if we ask:
What are the square roots of 49? The correct answer will be: The
numbers −7 or 7.
√
√
But if we ask: What is 49 =?. Then the correct answer is 49 = 7.
√
To represent the negative square root of a number a, we use − a.
√
Example: 100 = 10
√
Example: − 144 = −12
√
Example: −4 is not a real number.
1
Definition: A number r is the cube root of another number a if
r3 = a.
Example: 4 is the cube root of 64 since 43 = 64.
√
We use the symbol 3 a to represent the cube root of a.
√
3
Example: 343 = 7 since 73 = 343.
Notice that unlike square roots, any real number, even negative numbers, have cube roots, since a negative number to the third power is
negative.
√
Example: 3 −27 = −3
√
Example: 3 −1 = −1
Definition: A number r is a n-th root of a number a if rn = a.
Example: 2 is a 4-th root of 16 since 24 = 16.
Example: −2 is also a 4-th root of 16 since (−2)4 = 16.
Example: 5 is a 6-th root of 15625 since 56 = 15625.
Example: −3 is a 5-th root of −243 since (−3)5 = −243.
Notice that if n is an odd number, then any real number a has an n-th
root. If n is an even number, then only a positive real number a has
an n-th root.
If n is an even number, we define the principal n-th root of a positive
real number a as the positive number
whose n-th power is equal to a.
√
n
We represent this n-th root
of a by a. In other words, if a is positive
√
n
and n is even, then r = a if r > 0 and rn = a.
If n is an odd number, we still use the same symbol to represent the
n-th rood of a. If n is odd, the n-th root of any real number a exists,
and has the same sign as that of a.
√
4
Example: 81 = 3
√
Example: 3 −8 = −2
√
Example: 5 −243 = −3
√
4
Example: 256 = 4
2
We know that raising to a negative exponent means taking the reciprocal. Can we make sense of raise to a non-negative exponent? For
example, can 82/3 be defined?
1
We begin by trying to define raise a (positive) number to the power:
n
Observe that if we want the property of exponents
to hold, namely if
m n
mn
1/n n
we want (a ) = a , then we would want: a
= a(1/n)n = a1 = a.
In other words, a1/n raised to the n-th power should be equal to a.
We know that the n-th root of a has this property. In other words, we
should define
Definition: If a ≥ 0, then
√
a1/n = n a
√
3
Example: 81/3 = 8 = 2
√
√
2
Example: 251/2 = 25 = 25 = 5
√
Example: (−343)1/3 = 3 −343 = −7
Using properties of exponents, we see that am/n = a1/n
define
m
. So we
Definition: For any non-negative real number a,
√ m
√
am/n = n am = n a
√ 3
4
3/4
Example: 16 =
16 = 23 = 8
√
2
3
2/3
Example: 125 =
125 = 52 = 25
√
3
4
3/4
Example: (4x) =
4x
Example: Rewrite in exponent form:
q
5
2x2 y 3
Answer:
q
5
2x2 y 3 = 2x2 y 3
1/5
= 21/5 x2/5 y 3/5
The negative sign in an exponent still has the same meaning. So we
can raise a non-negative real number to a negative rational exponent:
3
Example: 27−2/3 =
1
1
1
1
=
=
=
√
2
3
272/3
32
9
27
Property of Radicals:
√
√ √
n
n
If a ≥ 0; b ≥ 0, then ab = n a b
√
s
n
a
a
n
If a ≥ 0; b > 0, then
= √
n
b
b
To simplify a radical expression, we want to move the largest perfect
n-th root out of the radical.
√
Example: Simplify: 98
√
√
√ √
√
Answer: 98 = 49 · 2 = 49 2 = 7 2
√
√
√
√
√
3
3
3
3
3
Example: 32 = 8 · 4 = 8 4 = 2 4
√
√
√
√
√
5
5
5
5
5
Example: 64x7 = 32 · 2 · x5 · x2 = 32x5 · 2x2 = 2x 2x2
v
√
√
u
u 27
3
27
3
Example: t = √ =
25
5
25
Radicals with the same n root and the same expression inside the
radical can also be added/subtracted as like terms:
√
√
√
Example: 4 5 + 7 5 = 11 5
√
√
Example: Simplify: 98 − 11 2
√
√
√
√
√
Answer: 98 − 11 2 = 7 2 − 11 2 = −4 2
√
√
Example: Simplify: 125x5 + 2x2 5x
√
√
√
√
√
Answer: 125x5 + 2x2 5x = 5x2 5x + 2x2 5x = 7x2 5x
We can also multiply radicals:
√ √ Example: Multiply: 3 + 6 5 − 3
√ √ √
√
√
Answer: 3 + 6 5 − 3 = 15 − 3 3 + 5 6 − 18
√
√
√
√
√
√
= 15 − 3 3 + 5 6 − 9 · 2 = 15 − 3 3 + 5 6 − 3 2
Sometime when a rational expression involves a radical in the denominator, we want to rationalize the denominator by removing the
radical in the denominator. We can sometimes do that by multiplying
4
the denominator by the appropriate factor:
3
Example: Rationalize the denominator: √
7
√
√
√
7
3
3
3 7
3 7
√ =√ ·√ =√ √ =
7
7
7
7
7 7
Example: Rationalize the denominator:
4
√
3+ 2
√
This one is a little tricker. We cannot just multiply by 2. Instead,
we multiply by the conjugate of the denominator:
√ √
√
4 3− 2
4
4
3− 2
12 − 4 2
√ =
√ ·
√ = √ √ =
=
9−2
3+ 2
3+ 2 3− 2
3+ 2 3− 2
√
12 − 4 2
7
To solve an equation that involves a radical, try to isolate the radical
expression, then raise to the appropriate exponent that is the reciproal
of the radical exponent:
Example: Solve the equation:
√
3
x=5
Answer: We raise both sides to the third power:
√ 3
3
x = (5)3
x = 125
Example: Solve the equation:
√
2x + 1 = 3
Answer: We square both sides:
√
2
2x + 1 = 32
2x + 1 = 9
2x = 8
x=4
5