Form 2 Unit 1 Lesson 4: – Third and Fourth Laws of Indices

Objective

At the end of the lesson, students should be able to use the third and the fourth index laws

When a number which is already expressed as an exponent is raised to another power the two indices are multiplied.

In general, if a is any number and m and n are positive integers, then

Third index law:

          (a^m)ⁿ⁼a^mn

          (a^m)ⁿ⁼a^mn

Example 1

Simplify (5²)³

Solution

(5²)³= 5² × 5² × 5²

= (5×5) × (5×5) × (5×5)

= 5×5×5×5×5×5

= 5⁶

Thus (5²)³=5^2×3=5⁶

Fourth index law:

Any number (except zero) raised to the power 0 is 1.

            a⁰ = 1 (a≠0)

If we write 2⁴ as the products, then we can see what happens when it is divided by itself.

Example 1

Simplify the following

a.(3⁶)⁴                                                            b.  (4³×3³)²

Solution  

a.(3⁶)⁴                                                 b.  (4³× 3³)²

=3^6×4                                                   = (4³)²× (3³)²

=3²⁴                                                        =4^3×2×3^3×2

                                                                               =4⁶×3⁶

Example 2

Simplify each of the following using the fourth index law

a. (5×6)⁰                      b. 8⁰                          

Solution

a.   (5×6)⁰                            b. 8⁰                           c. 2⁷ 2⁷

= 5⁰×6⁰                               =1                             =2^7-7

=1×1                                                                      =2⁰

=1                                                                           =1