Exponential notation

Some numbers are just long strings of the same factor multiplied over and over — the distance to a star in kilometres, the number of cells after many rounds of doubling, the population of a country. Writing out $5\times5\times5\times5$ every time is slow and error-prone. Exponential notation fixes this by compressing repeated multiplication of the same number into a short label: $5^4$ . This single idea opens the "Power Play" chapter and becomes the language for everything that follows in it — the laws of exponents, standard form, and comparing very large or very small numbers. Get comfortable moving both ways between the compact form and the expanded multiplication, and the rest of the chapter follows naturally.

The core insight

Exponential notation $a^n$ is shorthand for the base $a$ multiplied by itself $n$ times, nothing more and nothing less.

The core idea

If a number $a$ is multiplied by itself $n$ times, this repeated multiplication is written compactly as $a^n$ , read " $a$ raised to the power $n$ " or " $a$ to the $n$ th power":

a^n = \underbrace{a\times a\times a\times\cdots\times a}_{n\ \text{times}}

where: $a$ is the base (the number being repeated as a factor) and $n$ is the exponent, also called the power or index (the count of how many times the base appears). For example, $2^5 = 2\times2\times2\times2\times2 = 32$ : base $2$ , exponent $5$ , value $32$ . A few readings are special — $a^2$ is read " $a$ squared" (it gives the area of a square of side $a$ ) and $a^3$ is read " $a$ cubed" (the volume of a cube of side $a$ ). From $a^4$ onward we just say " $a$ to the power $4$ ", "to the power $5$ ", and so on. The base need not be a whole number — it can be a fraction, a decimal, or negative. Fluency means being able to go either way between the two forms:

3^4 = 3\times3\times3\times3 = 81 \qquad\qquad 81 = 3^4

It helps to be clear about what exponential notation is not. It is not a new arithmetic operation sitting beside addition and multiplication — it is purely a shorthand label for multiplication repeated a fixed number of times. Because of this, any exponential expression can always be unpacked back into ordinary multiplication; the notation only saves labour when the string of repeated factors would otherwise be very long. This is exactly why scientists reach for it to describe things like the number of stars in a galaxy or the number of bacteria after many divisions — writing $10^{11}$ is far more readable than typing out eleven zeros' worth of multiplication signs. Notice too how quickly the value grows compared to the exponent itself. Going from $2^1$ to $2^5$ the exponent only climbs by $4$ , yet the value climbs from $2$ all the way to $32$ — a sixteenfold jump. This fast growth is the whole reason exponential notation is useful for describing very large real-world counts: a population that doubles every generation, or a chain of cell divisions, reaches enormous numbers in surprisingly few steps, and writing $2^{20}$ is far easier to read, say, and compare than writing out the full product of twenty twos.

Figure

A ladder of increasing powers of two showing how each step multiplies the previous value by one more factor of two

Derivation AHow raising the exponent by one builds every power from the last

The goal is to show, using nothing but the definition of exponential notation as repeated multiplication, that going from $a^n$ to $a^{n+1}$ always means multiplying the previous value by exactly one more factor of the base, so that $a^{n+1} = a^n\times a$ , and to use this fact to build up a table of powers step by step instead of expanding each one from scratch every time.

Step 1. Start from the definition. By the meaning of exponential notation, $a^n$ is the base $a$ written out and multiplied by itself $n$ times: $a^n = \underbrace{a\times a\times\cdots\times a}_{n\ \text{times}}$ . Similarly $a^{n+1}$ is the base multiplied by itself $n+1$ times.

Step 2. Compare the two expansions. The expansion of $a^{n+1}$ is just the expansion of $a^n$ with one extra factor of $a$ attached at the end: $a^{n+1} = \underbrace{a\times a\times\cdots\times a}_{n\ \text{times}}\times a = a^n\times a$ . No new rule is being invented here — it is simply relabelling the same repeated multiplication.

Step 3. Use this to build powers of $2$ without re-expanding each time. Start at $2^1 = 2$ . Then $2^2 = 2^1\times 2 = 2\times2 = 4$ , then $2^3 = 2^2\times2 = 4\times2 = 8$ , then $2^4 = 2^3\times2 = 8\times2 = 16$ , then $2^5 = 2^4\times2 = 16\times2 = 32$ . Each new power is found from the one before it by a single multiplication, matching the figure above.

Step 4. Read the result. The relation $a^{n+1} = a^n\times a$ confirms that exponential notation behaves consistently as the exponent grows by one at a time — it is not a coincidence that powers of a number greater than $1$ keep growing faster and faster; each step multiplies in one more copy of the base. This step-by-step building block is exactly what later gets used to justify the laws of exponents.

Worked example

Write $6\times6\times6\times6\times6$ in exponential notation and find its value; then use the same base to evaluate $6^6$ .

Step 1. Count the repeated factor. The number $6$ appears $5$ times in the product, so the base is $6$ and the exponent is $5$ : $6\times6\times6\times6\times6 = 6^5$ .

Step 2. Evaluate by building up one power at a time, exactly as in the derivation above: $6^1=6,\ 6^2=36,\ 6^3=216,\ 6^4=1296,\ 6^5=7776$ . So $6^5 = 7776$ .

Step 3. Extend to $6^6$ using $a^{n+1}=a^n\times a$ instead of expanding six factors from scratch: $6^6 = 6^5\times6 = 7776\times6 = 46656$ .

Final answer: $6\times6\times6\times6\times6 = 6^5 = 7776$ , and $6^6 = 46656$ .

Competitive stretch

Expand $2^3\times3^2$ and find its value. Since the bases $2$ and $3$ are different, each term must be expanded separately before combining: $2^3 = 2\times2\times2 = 8$ and $3^2 = 3\times3 = 9$ , so $2^3\times3^2 = 8\times9 = 72$ . Checking by full expansion, $(2\times2\times2)\times(3\times3) = 72$ — this also matches the prime factorisation of $72$ , showing how exponential notation and prime factorisation connect.

Common traps

Confusing $a^n$ with $a\times n$ . $4^3$ means $4\times4\times4 = 64$ , not $4\times3=12$ ; the exponent counts repetitions of the base, it is not a second factor.
Swapping base and exponent. In $5^7$ the base is $5$ and the exponent is $7$ ; students sometimes read the sentence backwards when translating words into notation.
Assuming a negative base always gives a negative answer. $(-2)^4 = 16$ (positive, an even number of negative factors) while $(-2)^3=-8$ (negative, an odd number of factors) — the sign depends on whether the exponent is even or odd.
Trying to write different factors as one power. $2\times3\times4\times5$ cannot become a single $a^n$ because the factors are not all the same number; exponential notation only compresses a repeated factor.

Ask the tutor: Want to practise converting more expressions between expanded and exponential form, or check your understanding of the sign rule for negative bases? Ask the AI tutor for extra step-by-step examples.

Why does a negative base sometimes give a positive answer and sometimes negative?

Every time you multiply by a negative number, the sign of the running product flips. Starting from a positive $1$ , an even number of flips lands back on positive, and an odd number of flips lands on negative. So $(-2)^4$ flips sign four times and ends positive ( $16$ ), while $(-2)^3$ flips three times and ends negative ( $-8$ ). The quick rule: even exponent on a negative base gives a positive result, odd exponent gives a negative result.

Is 2 to the power 5 the same as 5 to the power 2?

No, and mixing these up is one of the most common slips. $2^5 = 2\times2\times2\times2\times2 = 32$ while $5^2 = 5\times5 = 25$ — different bases and different exponents give different values even though the same two digits are involved. Always check which number is being repeated (the base) and how many times (the exponent) before evaluating.

Do I always have to expand a power fully to find its value?

Not once you get comfortable — you can build it up one step at a time, the way the derivation above does: find $a^1$ , then multiply by $a$ again to get $a^2$ , multiply by $a$ again to get $a^3$ , and so on. This is faster and less error-prone than writing out a huge string of multiplication signs at once, especially for larger exponents, and it is exactly the pattern later used to justify the laws of exponents.

Exponential notation

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads