Rules of Exponents:
If n > 0, m > 0 are integers and x, y are any real numbers, then:
xm · xn = xm+n
xm
= xm−n
n
x
(xm )n = xmn
(xy)n = xn y n
n
x
xn
= n
y
y
Can we make sense of a negative exponent?
E.g.
x·x·x·x·x
x5
1
=
= 2
7
x
x·x·x·x·x·x·x x
xm
If this rule n = xm−n is to be true for m < n, then we must have
x
5
x
= x5−7 = x−2
7
x
1
This implies that we should define x−2 = 2
x
Definition of Negative Exponents:
If n is any integer and x 6= 0 is a real number, then
1
x−n = n
x
In other words, taking a negative expoenent is the same is taking the reciprocal
of the positive expoenent.
What about x0 ? If we want the rule to still hold, we have
x4
= x4−4 = x0
4
x
x4
but we know that 4 = 1, this naturally leads to:
x
Definition of zero exponent:
For all real number x, if x 6= 0, then x0 = 1
00 is undefined.
Simplify Expressions involving integer exponents:
In simplifying an expression involving exponents, remember that the Order of
Operation still holds and, in the absense of parenthesis, exponents have the
highest order of operation. This means that, in the absense of parenthesis, an
exponenet is applied only to the number/variable immediately below it.
E.g
1
1
=
2
4
16
1
1
−4−2 = − 2 = −
4
16
1
1
=
(−4)−2 =
(−4)2
16
1
3
3x−4 = 3 · 4 = 4
x
x
1
1
1
=
=
(3x)−4 =
(3x)4
34 x 4
81x4
−3
3
1
−3x−4 = −3 · 4 = 4 = − 4
x
x
x
1
1
1
=
=
(−3x)−4 =
(−3x)4
(−3)4 x4
81x4
4−2 =
30 = 1
−30 = −(30 ) = −(1) = −1
(−3)0 = 1
2x0 = 2(1) = 2
(2x)0 = 1
1
x
=
y3
y3
1
1
(xy)−3 =
= 3 3
3
(xy)
xy
1
1 44
5 − 3−2 = 5 − 2 = 5 − =
3
9
9
1
1
(5 − 3)−2 = 2−2 = 2 =
2
4
2
1
xy
1
xy 2 − 1
−2
x−y =x− 2 = 2 − 2 =
y
y
y
y2
1
1
(x − y)−2 =
=
(x − y)2
x2 − 2xy + y 2
xy −3 = x ·
(32 )−3 = 3−6 =
1
36
(x−2 )−4 = x8
Simplify:
x−3 x−5
Ans:
x−3 x−5 = x−3+−5 = x−8 =
1
x8
Simplify:
y −2
= y −2−(−6) = y −2+6 = y 4
−6
y
Simplify:
x−3 y 2
x2 y −1
One Approach:
x−3 y 2
1
y3
−3−2 2−(−1)
−5 3
3
=x
y
=x y = 5 ·y = 5
x2 y −1
x
x
Another Approach:
x−3 y 2
y2y1
y3
=
=
x2 y −1
x3 x2
x5
Simplify:
(x4 y 2 )3
Ans:
(x4 y 2 )3 = (x4 )3 (y 2 )3 = x12 y 6
Simplify:
2
x
y3
Ans:
2
x
x2
x2
= 3 2= 6
y3
(y )
y
Simplify:
1
3−2
Ans:
1
= 32 = 9
−2
3
Simplify:
a4 a3
b−2
Ans:
a4 a3
a7
= −2 = a7 b2
−2
b
b
Simplify:
−2
5
6
Ans:
−2 2
36
5
62
6
=
= 2=
6
5
5
25
Simplify:
2 3 2
4x y
3xy −2
Ans:
2 2 2
2 3 2 4 2−1 3−(−2)
4x y
4 5
4
=
x
y
xy
(x)2 (y 5 )2
=
=
−2
3xy
3
3
3
16 2 10 16x2 y 10
= xy =
9
9
Simplify:
−3 2 −3
x y
5x−1 y −3
Ans:
−3 2 −3 1 2 3 −3 5 −3 2 3
x y
xy y
y
5x
=
=
=
5x−1 y −3
5x3
5x2
y5
(5x2 )3
(5)3 (x2 )3
125x6
=
=
(y 5 )3
(y 5 )3
y 15
Simplify:
=
4y −3 y −10
−
y −2
Ans:
4y −3 y −10
4y −3+−10
4y −13
−
=−
= − −2
y −2
y −2
y
1
−4
= −4y −13−(−2) = −4y −11 = −4 · 11 = 11
y
y
Simplify:
−2
4x2 y 4 z −4
3x−2 y −1 z 2
Ans:
−2
2 4 −4 −2 4 2−(−2) 4−(−1) −4−2
4x y z
x
y
z
=
3x−2 y −1 z 2
3
−2 −2 4 5 −2
4 4 5 −6
4 4 51
4x y
xy z
xy 6
=
=
=
3
3
z
3z 6
2
(3)2 (z 6 )2
9z 12
(3z 6 )2
3z 6
=
=
=
=
4x4 y 5
(4x4 y 5 )2
(4)2 (x4 )2 (y 5 )2
16x8 y 10
Simplify:
−2
3(x−2 )−3
8(x−4 )−5
Ans:
−2 6 −2 −2 14 2
3x
3
8x
3(x−2 )−3
=
=
=
8(x−4 )−5
8x20
8x14
3
(8x14 )2
(8)2 (x14 )2
64x28
=
=
=
(3)2
9
9
Simplify:
(a3 b−4 )2 (2a4 b4 )−2
(a2 b6 )−3
Ans:
One Approach:
(a3 b−4 )2 (2a4 b4 )−2
(a3 )2 (b−4 )2 · (2)−2 (a4 )−2 (b4 )−2
=
(a2 b6 )−3
(a2 )−3 (b6 )−3
2−2 · a6+−8 b−8+−8
2−2 a−2 b−16
a6 b−8 · 2−2 a−8 b−8
=
=
=
a−6 b−18
a−6 b−18
a−6 b−18
1
1
a4 b 2
= 2−2 a−2−(−6) b−16−(−18) = 2−2 a4 b2 = 2 · a4 b2 = · a4 b2 =
2
4
4
Another Approach:
(a3 b−4 )2 (2a4 b4 )−2
(a3 b−4 )2 (a2 b6 )3
(a3 )2 (b−4 )2 · (a2 )3 (b6 )3
=
=
(a2 b6 )−3
(2a4 b4 )2
(2)2 (a4 )2 (b4 )2
a6 b−8 · a6 b18
a12 b10
1 12−8 10−8
=
=
=
·a
b
4a8 b8
4a8 b8
4
1
a4 b2
= · a4 b 2 =
4
4