Introduction to Exponents

What an exponent means

An exponent records how many times a number is used as a factor. In the power

the large number $2$ is the base (the number being multiplied). The small raised number $5$ is the exponent (it counts the copies of the base). The whole expression $2^5$ is called a power of $2$. You read it as “two to the fifth power,” or just “two to the fifth.”

In general, for a base $a$ and a whole-number exponent $n \ge 1$,

$a^n$ is the result of starting at $a$ and multiplying by $a$ again and again until $a$ has appeared as a factor $n$ times. The base $a$ can be any number you have met so far: a whole number, an integer, a fraction, or a decimal. (A negative base needs parentheses, as in $(-3)^4$; you’ll see why later in this lesson.) The exponent $n$ here is a whole number, and it is at least $1$.

The exponent is a count, not a thing you multiply by. So $2^5$ is not $2 \times 5 = 10$. The $5$ tells you how many $2$‘s to multiply, and five $2$‘s multiply to $32$. Read the exponent as an instruction (“use the base as a factor this many times”), never as a factor in its own right.

Reading powers aloud: squared and cubed

Most powers are read with the word “power,” as in ”$7$ to the fourth power.” Two exponents come up so often that they have their own names, and both names come from geometry.

A second power is read “squared.” A $4$ by $4$ square holds $4 \times 4 = 16$ unit squares, so “four squared is sixteen.” The name comes from that picture: $a^2$ is the area of a square whose side has length $a$.

A third power is read “cubed.” A $2$ by $2$ by $2$ cube is built from $2 \times 2 \times 2 = 8$ unit cubes, so “two cubed is eight.” Again the name comes from the shape: $a^3$ is the volume of a cube whose edge has length $a$. Past the third power there are no special words; you simply say “to the fourth,” “to the fifth,” and so on.

Evaluating powers

To evaluate a power means to work out the single number it equals. Write the base out the right number of times, then multiply step by step. Keep a running product as you go, so you never juggle more than two numbers at once.

The same procedure works when the base is a fraction or a decimal, because a power is still just repeated multiplication. To square a fraction you multiply it by itself, so $\left(\tfrac{1}{2}\right)^2 = \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}$, and to square a decimal you do the same, so $0.1^2 = 0.1 \times 0.1 = 0.01$.

Powers of ten and the digit-count pattern

Powers of ten deserve a closer look, because they are the backbone of place value and of writing large numbers. Watch what each one equals:

Every time the exponent goes up by one, the value gains one more zero.

This is why $10^6$ is a million ($1$ followed by six zeros) and $10^9$ is a billion, without writing out a single multiplication.

Zeros are not the same as digits. $10^5 = 100{,}000$ has five zeros, but six digits in total: the leading $1$ plus the five zeros. In general, $10^n$ has $n$ zeros and $n + 1$ digits. A number can have six digits without being a power of ten at all, like $300{,}000$; only a $1$ followed entirely by zeros counts.

How fast does a power grow?

The reason exponents matter so much is the speed, but only when the base is bigger than $1$. Repeated addition climbs in even steps, while repeated multiplication by a number bigger than $1$ accelerates: each step multiplies by that same number again, so the jumps keep getting bigger. Watch what happens as the exponent on $2$ climbs by one each time:

Each row doubles the one above it. By $2^{10}$ you have already passed a thousand ($2^{10} = 1024$), starting from a single $2$. That runaway growth shows up anywhere something doubles: a sheet of paper folded again and again, a savings balance that doubles each year, or a rumor that spreads to two new people at each step.

Not every base grows this way. A base of $1$ stays at $1$ no matter the exponent ($1^{100} = 1$), and a positive fraction less than $1$ actually shrinks: $\left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4}$ is smaller than $\tfrac{1}{2}$.

Order of operations with exponents

When a power sits inside a longer expression, where does it fall in the order of operations? Evaluate the power first, then multiply or divide, and finally add or subtract.

So powers are evaluated after parentheses but before multiplication, division, addition, and subtraction. This is the “E” in the mnemonic PEMDAS.

Parentheses override this, because you work inside them first of all. So $(2 + 3)^2$ adds before squaring, giving $5^2 = 25$, a different number from $2 + 3^2 = 11$. Whenever you want the exponent to apply to a whole sum or product, you must wrap that sum or product in parentheses.

The crucial case: a minus sign and a square

The place this matters most, and the place almost everyone slips at least once, is a minus sign next to a power. Compare these two expressions carefully; they look almost identical but give opposite signs:

In $(-2)^2$, the parentheses make $-2$ the base, so the whole negative number is squared. In $-2^2$, there are no parentheses, so the exponent applies only to the $2$, and the minus sign applies afterward.

The same logic settles odd powers, where the canceling works out differently. In $(-2)^3$ you multiply three negative factors, and three minus signs leave the result negative: $(-2)^3 = (-2)(-2)(-2) = -8$. Compare $-2^3$: with no parentheses the exponent applies only to $2$, so $-2^3 = -(2 \times 2 \times 2) = -8$. Here the two forms happen to agree, but only because the exponent is odd. For an even exponent, like the square above, they disagree: $(-2)^2 = 4$ but $-2^2 = -4$. Count the negative factors of a nonzero base: an even count cancels to a positive, an odd count stays negative.

Common mistakes