Integer Exponents

The nth power of a real number k is obtained by multiplying the number k by itself n times $$ k^n = \underbrace{k \cdot \ k \cdot k \cdot ... \cdot k}_{n \ times} $$ The real number k is called the base, while the integer n is called the exponent.

For example, the power 4³ is obtained by multiplying the number 4 by itself three times. It is read as "four to the third power" or "four cubed".

$$ 4^3 = 4 \cdot 4 \cdot 4 = 64 $$

Instead of writing the factor 4 several times, we use the exponential notation 4³.

A power with an integer exponent is defined for every real number k when the exponent is a positive integer n>0.

If the exponent is positive n>0, the power kⁿ is the product of n factors equal to the base k. $$ k^n = \underbrace{k \cdot \ k \cdot k \cdot ... \cdot k}_{n \ times} $$

The expression is not defined when both the base and the exponent are zero (0⁰), because this case has no defined value.

Note. If the exponent is n=2, the power is called the square, because the value k² corresponds to the area of a square whose side length is k. If the exponent is n=3, the power is called the cube, because the value k³ corresponds to the volume of a cube whose edge length is k.

If the exponent n is equal to one k¹, the power is equal to the base itself k.

$$ k^1 = k $$

For example, the power 4¹ is equal to 4, and the same holds for any number.

$$ 1^1=1 \\ 2^1=2 \\ 3^1=3 \\ 4^1=4 \\ 5^1=5 \\ \vdots $$

By convention, a power with exponent zero k⁰ is equal to 1 for any nonzero base (k≠0).

$$ k^0 = 1 \ \ \ \ \forall \ k \ne 0 $$

For example

$$ 1^0=1 \\ 2^0=1 \\ 3^0=1 \\ 4^0=1 \\ 5^0=1 \\ \vdots $$

On a separate page I provide an example that explains why any number raised to the power of zero equals one.

Note. The expression 0⁰, that is zero raised to the power of zero, has no defined value and is considered undefined.

If the exponent is a negative integer k^-n, the power is defined as the reciprocal of the corresponding positive power.

$$ k^{-n} = \frac{1}{k^n} $$

For example, the power 4^-3 is the reciprocal of 4³.

$$ 4^{-3} = \frac{1}{4^3} = \frac{1}{4 \cdot 4 \cdot 4} = \frac{1}{64} $$

Useful observations

Here are some useful observations about powers with integer exponents.

• If the base is positive k>0, the power kⁿ is always positive.
• If the base is positive and greater than one (k>1), the powers kⁿ increase as the exponent n increases.
  
• If the base is positive and between zero and one (0<k<1), the powers kⁿ decrease as the exponent n increases.
  
• If the base is equal to one (k=1), the power kⁿ is always equal to one for any exponent n.
  
• If the base is negative (k<0), the powers kⁿ are positive when the exponent n is even and negative when the exponent n is odd. Therefore successive powers alternate in sign. For example $$ (-2)^1 = -2 \\ (-2)^2 = +4 \\ (-2)^3 = -8 \\ (-2)^4 = +16 \\ (-2)^5 = -32 \\ \vdots $$

  Note. Powers with a negative base have increasing absolute values |kⁿ| as the exponent n increases when the absolute value of the base satisfies |k|>1. For example $$ |(-2)^1| = |-2|=2 \\ |(-2)^2| = |4|=4 \\ |(-2)^3| = |-8| = 8 \\ |(-2)^4| = |16| = 16 \\ |(-2)^5| = |-32|=32 \\ \vdots $$ The absolute values |kⁿ| decrease instead when the absolute value of the base satisfies 0<|k|<1. $$ |(- \frac{1}{2})^1| = |-\frac{1}{2}|=\frac{1}{2} \\ |(-\frac{1}{2})^2| = |\frac{1}{4}|=\frac{1}{4} \\ |(-\frac{1}{2})^3| = |-\frac{1}{8}| = \frac{1}{8} \\ |(-\frac{1}{2})^4| = |\frac{1}{16}| = \frac{1}{16} \\ |(-\frac{1}{2})^5| = |-\frac{1}{32}|=\frac{1}{32} \\ \vdots $$

And so on.