Index Laws in Algebra

Every time a file is measured in gigabytes, a scientist writes a huge distance as 10^{9} metres, or a population doubles and doubles again, powers are being multiplied and divided behind the scenes. The index laws are the handful of shortcuts that make working with those powers quick — and they work just as smoothly with letters as with numbers.

You have already met the index laws with numbers: to work out 2^3 \times 2^4 you don't reach for a calculator, you just add the indices — 3 + 4 = 7, so the answer is 2^7. Here is the good news: those exact same laws work with letters, and they don't care what the letter stands for.

Swap the 2 for an x and nothing else changes:

x^3 \times x^4 = x^{7}

That tiny move — from a fixed number to a stand-in letter — is what makes index laws the workhorse of algebra. Multiplying out polynomials, tidying up a formula, or getting a function into a shape you can graph all lean on adding and subtracting indices. Learn it once with letters and it powers everything downstream.

The reason is the same one you saw with numbers: a power is just shorthand for repeated multiplication, so x^3 means x \times x \times x. Multiply two powers of the same base and you only have to count the factors. Press play to watch them line up and collapse.

The three laws, in letters

The counting argument gives all three index laws at once. When you multiply powers of one base, add the indices:

x^m \times x^n = x^{m+n}

When you divide, every factor on the bottom cancels one on top, so you subtract the indices:

\frac{x^m}{x^n} = x^{m-n}

And a power of a power means you have n copies of x^m — that is m factors repeated n times — so you multiply the indices:

(x^m)^n = x^{mn}

These only work when the base is the same letter. Powers of different bases — say x^2 \times y^3 — cannot be merged, just as you cannot combine unlike terms when collecting like terms.

What about the numbers in front?

Real algebra rarely gives you a bare x^3. It gives you 2x^3 — a coefficient (the number in front) stuck onto a power. When you multiply these, you handle the two parts separately:

2x^3 \times 3x^4 = (2 \times 3)\,(x^3 \times x^4) = 6x^{7}

The coefficients multiply like ordinary numbers (2 \times 3 = 6); the indices add (3 + 4 = 7). Two different jobs, done side by side. Division works the same way — divide the numbers, subtract the indices.

Worked examples

1. A product. Simplify 2x^3 \times 5x^2.

2x^3 \times 5x^2 = (2 \times 5)\,x^{3+2} = 10x^{5}

Multiply the numbers 2 \times 5 = 10; add the indices 3 + 2 = 5.

2. A quotient. Simplify 12x^5 \div 4x^2.

\frac{12x^5}{4x^2} = \frac{12}{4}\,x^{5-2} = 3x^{3}

Divide the numbers 12 \div 4 = 3; subtract the indices 5 - 2 = 3.

3. A power of a power. Simplify (3x^2)^3.

(3x^2)^3 = 3^3 \times (x^2)^3 = 27x^{6}

Everything inside the bracket is cubed. The 3 becomes 3^3 = 27, and the index is multiplied 2 \times 3 = 6.

4. A mix. Simplify \dfrac{(2x^3)^2 \times 3x}{4x^4}.

\frac{(2x^3)^2 \times 3x}{4x^4} = \frac{4x^6 \times 3x}{4x^4} = \frac{12x^{7}}{4x^4} = 3x^{3}

First expand the bracket ((2x^3)^2 = 4x^6), then multiply on top (add indices: 6 + 1 = 7), then divide (subtract: 7 - 4 = 3). One law at a time and it falls out cleanly.

The single most common slip is blurring the numbers and the indices into one operation. They get treated differently:

This little "add the indices" rule is quietly everywhere. In calculus you differentiate and integrate powers by nudging the index up or down — impossible without it. It runs the exponential functions behind population growth, radioactive decay, and the compound interest ticking up in a savings account.

And here is the deepest twist: the rule x^a \times x^b = x^{a+b} says that multiplying powers turns into adding their indices. That single fact is the whole reason logarithms exist — a logarithm is just an index, and it converts a hard multiplication into an easy addition. Before calculators, that trick let engineers multiply enormous numbers with a ruler (the slide rule). All from adding indices.