Properties of Integer Exponents

Question 1

For n and m integers and a and b real numbers we have:

The Product Rule:

a^n.a^m = ?

Answer 1

The Product Rule:

a^n.a^m = a^n+m

e.g. 3^2.3^3 = 3^2+3 = 3^5

Question 2

The Power of a Power Rule:

(a^n)^m = ?

Answer 2

The Power of a Power Rule:

(a^n)^m = a^n.m

e.g. (a^2)^3 =a^2.3 = a^6

Question 3

The Power of a Product Rule:

(ab)^m = ?

Answer 3

The Power of a Product Rule:

(ab) ^m=a^m.b^m
e. g. (ab)^4 = a^4.b^4

Question 4

The Power of a Quotient Rule:

(a/b)^m = ?

Answer 4

The Power of a Quotient Rule:

(a/b)^m = a^m/b^m

Where b does not = 0

e.g. (a/b)^3 = a^3/b^3

Question 5

The Quotient Rule:

a^m/a^n = ?

Answer 5

The Quotient Rule:

a^m/a^n = a^m-n

e.g. a^4/a^2 = a^4-2= a^2

Question 6

The Zero Exponent Rule:

a^0 = 1, when a is not equal to 0

Answer 6

The Zero Exponent Rule:

a^0 = 1, when a is not equal to 0

e.g. 5^0 = 1

Question 7

The Negative Exponent Rule:

a^-n = ?

Answer 7

The Negative Exponent Rule:

a^-n = 1/a^n

a^-4= 1/a^4

Question 8

Ex. Simplify the following expression and write your answer using only positive exponents.

(2w^-5 v^-6) (3u^7 u^2) (5v^7 w^4)

Answer 8

Since multiplication is both commutative and associative we can regroup this multiplication as follows:

=(2.3.5)(w^-5.w^4)(v^-6.v^7)(u^7.u^2)
=30.w^-5+4.v^-6+7.u^7+2
=30.w^-1.v^1.u^9
= 30 1/w^1 v^1 u^9

(drop exponents of ^1 as we don’t need to specify them e.g 2^1 is just 2)
= 30.v.u^9/w

Question 9

Ex. Simplify the following expression and write your answer using only positive exponents.

(6mn^-2/3n2m-1)^-3

Answer 9

Step 1: group like terms together again

= (6/3 . m/m^-1 . n^-2/n^2)^-3
= (2 . m^1-(-1) . n^-2-2)^-3
= (2 . m^2 . n^-4)^-3
= then by the power of the product rule you can raise each factor to the power of -3

(2)^-3 (m^2) ^-3 (n^-4)^-3

= then by the negative exponent rule

1/2^3 . m^-6 . n^12

= 1/8 . 1/m^6 . n^12
= n^12/8m^6