Exponents and Powers - Exponents and their Meaning

Exponents are a shorthand way to represent repeated multiplication of the same number. Instead of writing $2 \times 2 \times 2 \times 2$, we write it as $2^4$. Visually, the base is written as a normal-sized number, while the exponent is written as a smaller superscript at the top-right corner of the base.

In the expression $a^n$, $a$ is known as the base (the number that is being multiplied) and $n$ is known as the exponent or index (representing how many times the base appears in the multiplication). The entire expression $a^n$ is read as '$a$ raised to the power $n$'.

Expanded form refers to writing the multiplication in full. For example, $5^3$ in expanded form is $5 \times 5 \times 5$. Exponential form is the condensed version. You can visualize this like an accordion that can be stretched out (expanded form) or squeezed together (exponential form).

Powers of negative numbers depend on whether the exponent is even or odd. If the exponent is an even number, the product will be positive because negative signs pair up and cancel out, such as $(-3)^2 = 9$. If the exponent is an odd number, the product will be negative, such as $(-3)^3 = -27$.

Any non-zero number raised to the power of zero is always $1$. For example, $15^0 = 1$ and $x^0 = 1$. This can be understood by looking at the pattern of decreasing powers: as you divide the base by itself, you eventually reach the value of $1$.

The number $1$ raised to any power is always $1$ ($1^{100} = 1$). Similarly, the number $-1$ raised to an even power is $1$, and $-1$ raised to an odd power is $-1$. This acts like a 'toggle' switch in calculations.

Squaring a number means raising it to the power of $2$, such as $a^2$, which represents the area of a square with side length $a$. Cubing a number means raising it to the power of $3$, such as $a^3$, representing the volume of a cube with side length $a$.