Imagine you need to multiply a number by itself several times. For example, $5 \times 5 \times 5 \times 5$. Writing this out can get tedious, especially if you have to do it many times.

Indices, also known as exponents or powers, are a shorthand for this exact situation. Instead of writing $5 \times 5 \times 5 \times 5$, we can simply write $5^4$. This is read as "five to the power of four."

The index tells you how many times to multiply the base by itself. So, $10^2$ means $10 \times 10$, which is 100. And $2^5$ means $2 \times 2 \times 2 \times 2 \times 2$, which is 32. This simple notation is the key to handling very large and very small numbers.

To work with indices efficiently, we use a few fundamental rules. These rules aren't arbitrary; they come directly from the definition of what an index is.

Multiplying Powers

What happens when you multiply two powers that have the same base, like $x^2 \cdot x^3$? Let's write it out:

$(x \cdot x) \cdot (x \cdot x \cdot x) = x \cdot x \cdot x \cdot x \cdot x = x^5$

Notice that the new index, 5, is just the sum of the original indices, 2 and 3. This gives us our first law.

Dividing Powers

Now let's try division. What is $x^5 \div x^2$? Again, we can expand it:

$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$

Two of the $x$'s on top cancel out with the two on the bottom, leaving us with $x \cdot x \cdot x$, or $x^3$. The new index, 3, is the difference between the original indices, 5 and 2. This is our second law.

A Power of a Power

Finally, what if you raise a power to another power, like $(x^2)^3$? This means we multiply $x^2$ by itself three times:

$x^2 \cdot x^2 \cdot x^2 = x^{2+2+2} = x^6$

The resulting index, 6, is the product of the original indices, 2 and 3. This leads to our third law.

These laws also help us understand what happens with some less obvious indices, like zero, negative numbers, and fractions.

The Zero Index

What is anything to the power of zero? Let's use the division law. We know that any number divided by itself is 1. So, $x^3 \div x^3 = 1$.

But according to the division law, $x^3 \div x^3 = x^{3-3} = x^0$. Since both are true, it must be that:

This holds true for any base $a$ that is not zero.

Negative Indices

What about a negative power, like $2^{-3}$? Again, we can turn to the division law. Let's look at $x^2 \div x^5$.

Using the law, we get $x^{2-5} = x^{-3}$.

If we write it out the long way, we get:

$\frac{x \cdot x}{x \cdot x \cdot x \cdot x \cdot x} = \frac{1}{x \cdot x \cdot x} = \frac{1}{x^3}$

So, a negative index simply means taking the reciprocal of the base raised to the positive index.

This is incredibly useful for writing very small numbers. For example, $0.001$ is $\frac{1}{1000}$, which can be written as $\frac{1}{10^3}$, or simply $10^{-3}$.

Fractional Indices

What does a fraction in the index mean? A fractional index like $\frac{1}{n}$ indicates an $n$-th root. For example, $x^{1/2}$ is the square root of $x$, and $x^{1/3}$ is the cube root of $x$.

This idea combines with our other rules. What is $8^{2/3}$? We can think of this as $(8^{1/3})^2$. First, we take the cube root of 8, which is 2. Then, we square it:

$8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$

Understanding these rules allows us to simplify complex expressions into much neater forms.