When we talk about powers of integers, there are a number of principles that must be followed:

When the base of the power is any nonzero integer and the value of the exponent is an even number, the result of the power will be a positive integer.

**For example.**

- $(4) ^ 2 = 4 \cdot 4 = 16$

When the base of the power is a positive number and the value of the exponent is any integer, the result of the powering will also be positive.

** For example:**

- $(4) ^ 3 = 4 \cdot 4 \cdot 4 = 64$

When the base of the power is a negative number and the value of the exponent is an even number, the result of the potentiation will be positive.

** For example:**

- $(- 4) ^ 2 = (-4) \cdot (-4) = 16$

When the base of the power is a negative number and the value of the exponent is an odd number, the result of the potentiation will be negative.

** For example:**

- $(-4)^3 = (-4) \cdot (-4) \cdot (-4) =-64$

**What is the powering of an integer number**? **$A$**** to the** **$n$****?**

It is the multiplication of $A$ by itself $n$ times, i.e. $A\times A\times A\times..\ldots\times A$.

**What is the sign of a power whose base is a negative integer and the exponent is an odd number?**

The sign is negative. Example: $(-2)^5=(-2)\times(-2)\times(-2)\times(-2)\times(-2)=-32$.

**What is the sign of a power whose base is any negative integer and the exponent is an even number?**

The sign is negative, since if it is a number $A$ is an integer and $n$ is an even number, we have to $n=2K$, where $k$ is an integer, so $A^n=A^(2k)=(A^k)^2$, which is the square of an integer and this is always greater than or equal to zero.

**Examples:**

a) $9^3=9x9x9=729$

b) $(-2)^3=(-2)x(-2)x(-2)$

c) $(-1)^11=-1$

d) $10^3=10x10x10=1000$

e) $12^3=12x12x12=1728$

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