Integer powering

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:

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

• $2^3=2\cdot2\cdot2=8$
• $3^3=3\cdot3\cdot3=27$
• $4^3=4\cdot4\cdot4=64$
• $5^3=5\cdot5\cdot5=125$
• $(-3)^2=(-3)\cdot(-3)=9$
• $(-5)^2=(-5)\cdot(-5)=25$
• $(-2)^3=(-2)\cdot(-2)\cdot(-2)=-8$
• $(-7)^3=(-7)\cdot(-7)\cdot(-7)=-343$
• $(-6)^3=(-6)\cdot(-6)\cdot(-6)=-216$

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) $12^3=12x12x12=1728$

If you liked this article, you may also be interested in the following articles

• The exponent of a power
• Base of a power
• Multiplication of powers of equal base
• Division of powers of equal base
• Power of a multiplication
• Power of a quotient
• Power of a power
• Power with zero exponent
• Powers of negative integer exponent
• Taking advantage of all the properties of powers or exponent laws
• Powering integers

On the Tutorela blog you will find a variety of articles about mathematics.