Exponent and Radicals - Rules for Manipulation
Algebraic Rules for Manipulating Exponential and Radicals Expressions.
In the following, n, m, k, j are arbitrary .
they can be integers or rationals or real numbers.
bn · bm
= bn+m−k
k
b
an · bm
ck
j
=
an·j · bm·j
ck·j
Add exponents in the numerator and
Subtract exponent in denominator.
The exponent outside the parentheses
Multiplies the exponents inside.
an
bm
b0 = 1
−1
bm
= n
a
b = b1
Negative exponent ”flips” a fraction.
Don’t forget these
Convert Radicals to Exponent notation
√
a = a1/2
√
m
a = a1/m
√
m
an = an/m
Radicals - Reducing
√
√
2·b=a b
a
√
√
m
am · b = a m b
Remove squares from inside
Exponent and Radicals - Solving Equations
Solve a power by a root
n/m
m/n
x
=y⇔x=y
Solve a root by a power
1
Example
3
2
a) Simplify
5
3
2·2·2
8
2
23
=
= 3 =
5
5
5·5·5
125
Method
2
2 · 32
b) Simplify
53
2
4 · 81
324
2 · 32
22 · 32·2
Method
=
=
=
3
3·2
5
5
15, 625
15, 625
Illustration: where is the negative?
c) Simplify
−3
Method
d) Simplify − 3
Method
4
( the ’negative’ is inside the parentheses)
−3
4
4
= (−3) · (−3) · (−3) · (−3) = 81
( the ’negative’ is outside the parentheses)
− 3
4
= −(3) · (3) · (3) · (3) = −81
−3
2
( the ’negative’ is in the exponent)
e) Simplify
5
−3
3
2
1
5
53
Method
=
=
(or
=
)
=
5
(2/5)3
3
23
=
125
8
2
More Examples
f) Simplify
Method
x3 · x7
x5
x3 · x7
= x3+7−5 = x5
5
x
g) Simplify (2a3 b2 )(3ab4 )3
Method
(2a3 b2 )(3ab4 )3 = 2a3 b2 · 33 a3 b4·3
= (2 · 27)(a3+3 )(b2+12 )
= 54a6 b14
3 2 4
y x
x
(give answer with only positive exponents )
h) Simplify
y
z
3 2 4
y x
x3 y 2·4 x4
x
Method
= 3·
y
z
y
z4
x3+4 y 8−3
x7 y 5
=
=
z4
z4
3
More Examples with negatives
i) Simplify
6st−4
(give answer with only positive exponents )
2s−2 t2
Negative exponents flip location: A negative exponent in the numerator
moves to the denominator. And a negative exponent in the denominator
moves to the numerator.
6ss2
3s3
6st−4
= 42 = 6
2s−2 t2
2t t
t
Method
j) Simplify
y
3z 3
−2
(give answer with only positive exponents )
A Negative exponent ’flips’ the fraction.
Method
y
3z 3
−2
=
3z 3
y
4
2
9z6
=
y2
More Examples
k) Simplify
Method
(2x3 )2 (3x4 )
(x3 )4
(2x3 )2 (3x4 ) 22 x3·2 · 3x4
=
(x3 )4
x3·4
(4 · 3)x6 x4
12
6+4−12
=
=
12x
=
x12
x2
5
Examples Simplifying Roots
a) Simplify
√
8
√
Method
b) Simplify
√
Method
√
3
√
√
4·2=2 2
√
√
25 · 3 = 5 3
75
√
c) Simplify
8=
75 =
x4
√
3
Method
x4 =
√
3
√
x3 · x = x 3 x
p
d) Simplify 4 81x8 y 4
Method
p
p
√
√
4
4
4
81x8 y 4 = 81 x8 4 y 4 = 3x2 y
√
√
4
Digression: Technically x2 = |x| and x4 = |x| but we will not
worry about that at this time.
6
More Examples
√
e) Simplify
32 +
√
√
Method
√
f) Simplify
25b −
√
Method
200
32 +
√
√
√
16 · 2 + 100 · 2
√
√
√
= 4 2 + 10 2 = 14 2
200 =
√
b3
25b −
√
√
25 · b + b2 · b
√
√
√
= 5 b − b b = (5 − b) b
b3 =
√
7