Exponents

$2 \times 2 \times 2 \times 2$ is tedious to write, so mathematicians abbreviate it to $2^4$ . Do not underestimate this shortcut: Multiplication compresses repeated addition, and exponents compress repeated multiplication. Imagine folding a sheet of paper — 2 layers after one fold, 4 after two … ten folds make $2^{10} = 1024$ layers.

Base and exponent: shorthand for repeated multiplication

$a^n$ reads "a to the power of n": $a$ is the base, $n$ is the exponent, and the meaning is n copies of a multiplied together.

$2^4 = 2 \times 2 \times 2 \times 2 = 16$
$10^3 = 10 \times 10 \times 10 = 1000$

Careful: $2^4 \neq 2 \times 4$ . The exponent says "multiply this many times", not "multiply by this".

Is the minus sign inside the brackets?

$(-2)^2 = (-2) \times (-2) = 4$ : the sign is inside, so it gets squared too. $-2^2 = -(2^2) = -4$ : the sign sits outside, so you square first and negate after. One pair of brackets, one flipped sign — the same story as substituting $x = -2$ into $x^2$ (see Substitution).

Three laws of exponents

When powers with the same base meet, the exponents follow clear rules:

Operation	Law	Numeric check
Same-base multiply	$a^m \times a^n = a^{m+n}$	$2^2 \times 2^3 = 4 \times 8 = 32 = 2^5$ ✓
Same-base divide	$a^m \div a^n = a^{m-n}$	$2^5 \div 2^2 = 32 \div 4 = 8 = 2^3$ ✓
Power of a power	$(a^m)^n = a^{mn}$	$(3^2)^2 = 9^2 = 81 = 3^4$ ✓

The reason is easy to see: $2^2 \times 2^3$ is two 2s multiplied by three more 2s — five 2s in total, so the exponents add.

Verify a law yourself

Compute $(10^2)^3$ the honest way: $10^2 = 100$ , then $100 \times 100 \times 100 = 1000000$ . By the law: multiply the exponents, $2 \times 3 = 6$ , giving $10^6 = 1000000$ . Two roads, one answer.

Why the zeroth power is 1

$a^0 = 1$ is not an arbitrary decree — watch the pattern:

$2^4 = 16, \quad 2^3 = 8, \quad 2^2 = 4, \quad 2^1 = 2$

Each step right lowers the exponent by 1 and divides the value by 2. Follow the rhythm to the end: $2^0 = 2 \div 2 = 1$ . The same reasoning works for any base, as long as $a \neq 0$ : $a^0 = 1$ .

Squares, cubes, and numbers too big to write

Two names worth knowing: $a^2$ is "a squared", because a square's area is the side length squared; $a^3$ is "a cubed", because a cube's volume is the edge length cubed. Higher powers are simply "a to the n-th".

Exponents truly shine with big numbers. Place Value showed how powers of ten position every digit; scientific notation goes one step further, squeezing any big number into "a number times a power of ten".

How fast is light?

Light travels about 300000000 metres per second — a 3 followed by eight zeros. In scientific notation that is $3 \times 10^8$ m/s: shift the decimal point eight places to the right and the original number comes back. Far easier to write — and to read — than a string of zeros.

Check yourself

Quick quiz

1. What is 2³×2²?
2. What is 5⁰?
3. What are (-2)² and −2²?