Åk 6–9

4.4 Powers
In section 4.2 you may have become acquainted with calculating with powers and here we will work a bit more with algebraic exponents.  Multiplication can be written as a power.

• the product of a number: 7 · 7 · 7 · 7 = 74

• the product of a variable: x · x · x · x = x4

You know that there is a difference between addition and multiplication.

Multiplication is a shortened addition of the same terms
2r + 2r = r + r + r + r = 4 · r = 4r

Powers can be seen as multiplication of similar factos.
2r · 2r = 2 · r · 2 · r = 2 · 2 · r · r = 4 · r2 = 4r2

Powers with positive bases
A. How can we simplify: (2x4)3?

(2x4)3 = 2x4 · 2x4 · 2x4 = 2 · 2 · 2 · x4 · x4 · x4 = 8 · x3 · 4 = 8x12

B. How can we simplify: (3b2c)3?

(3b2c)3 = (3b2c)(3b2c)(3b2c) = 3 · 3 · 3 · b2 · b2 · b2 · c · c · c =

= 27b2·3c1·3 = 27b6c3  Powers with a negative base
You know from before the calculation rules for multiplication with negative numbers.

A. (-a2c)3 = (-a2c) · (-a2c) · (-a2c) = -a2·3c1·3 = -a6c3

B. (-2bd)4 = (-2bd) · (-2bd) · (-2bd) · (-2bd) = 16 · b4 · d4 = 16b4d4

C. (-abc)6 = a6b6c6  