2. Introduction
an = a * a * a * a…* a (where there are n factors)
The number ‘a’ is the base and ‘n’ is the exponent.
For any non-zero rational number a, we define a0 =
1
Let a be any non-zero rational number and n be a
positive integer, then we define a-n = 1/an
3. Laws of Exponents
aman = am+n
When multiplying two powers of the same base,
add the exponents.
am/ an = am – n
When dividing two powers of the same base,
subtract the exponents.
(am)n = amn
When raising a power to a power, multiply the
exponents.
4. Laws of Exponents
(ab)n = anbn
When raising a product to a power, raise each factor to the
power.
(a/b)n = an / bn
When raising a quotient to a power, raise both the numerator
and denominator to the power.
(a/b)-n = (b/a)n
When raising a quotient to a negative power, take the
reciprocal and change the power to a positive.
a-m / b-n = bm / an
To simplify a negative exponent, move it to the opposite
position in the fraction. The exponent then becomes
positive.
5. nth root
If n is any positive integer, then the principal nth root of a is defined
as:
If n is even, a and b must be positive.
means nn
a b b a
6. Properties of nth roots
if n is odd
| | if n is even
n n n
n
n
n
m n mn
n n
n n
ab a b
a a
b b
a a
a a
a a
7. Rational Exponents
For any rational exponent m/n in lowest terms, where m
and n are integers and n>0, we define:
If n is even, then we require that a ≥ 0.
am is called the radicand
n is called the index
is called the Radical
In general, am/n=(am)1/n
In general, a-m/n= 1/am/n =1/(am)1/n
/ nm n m
a a
/ nm n m
a a
8. Examples
Express in exponential form:
• √7
SOLUTION:
√7 = 7 ½
Express in radical form:
• (5) 1/3
SOLUTION:
(5) 1/3 =
𝟑
𝟓
9. Examples
Find the value of
• (125) 2/3
SOLUTION:
(125) 2/3
= (
𝟑
𝟏𝟐𝟓)2
= 52
= 25
10. Examples
Find the value of
• (27)-2/3
SOLUTION:
(27) -2/3
= (
𝟑
𝟐𝟕)-2
= 3-2
= 1/9
11. Examples
Find the value of
• (6) ½ x (6) 3/2
SOLUTION:
(6) ½ + 3/2
= 6 4/2
= 62
= 36