- exponents

exponentiations

Exponentiation is written as bⁿ, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors, each of which is equal to b (the product itself can also be called power):

$b^n = \underbrace{b \times \cdots \times b}_n$

b⁰ = 1

b¹ = a

example

3 ⁵ = 3 · 3 · 3 · 3 · 3 = 243

exercise

3²	0¹⁵
2³	(-1)¹⁸
(3²)³	(-1)¹³
(-5)²	(-18)³	9⁰
	(1)¹³

answer

multiplaying powers

powers with same exponents

a and b two non zero rationnals

m a positif integer

aⁿx bⁿ = ( axb)ⁿ

powers with same bases

a is non zero rationnal

m and n two positif integers

a^mx aⁿ = a^m+n

the sign of power

* the power is negative when the base is negative

and the exponent in an odd number

* the power is positive when the base is negative

and the exponent in an even number

* the power is positive when the base is positive

when the base is power

x is a positive non zero integer

m and n two positif integers

$\ (x^m)^n=x^{mn}$

example

symplify

1) (2a⁴)x(5a⁷)

the answer ◄

2) (2a⁵)³

the answer ◄

exercises

1) prove that

81⁴ = 9⁸
32¹² = 2⁶⁰.

2) find the negative powers

3⁷ ; (-3)¹⁷ ; (-3)¹² ; -3⁹ ; -3¹²

the answer