An exponent is repeated multiplication — 2^10 is 1,024 — and the two extensions people find awkward both follow from one principle. A negative exponent gives the reciprocal, so 2^−1 is 0.5. A fractional exponent gives a root, so 9^0.5 is 3. The nth root of x is simply x raised to 1/n, which means roots and powers are the same operation rather than two.
Three rules then cover nearly all the arithmetic.
What are the three rules?
They come from counting the multiplications, which is the reason they are worth understanding rather than memorising.
| Rule | Form | Example |
|---|---|---|
| Multiply, same base | add exponents | 2³ × 2⁴ = 2⁷ |
| Divide, same base | subtract exponents | 2⁷ ÷ 2⁴ = 2³ |
| Power of a power | multiply exponents | (2³)⁴ = 2¹² |
The negative exponent falls straight out of the second rule. 2³ ÷ 2³ is 2⁰, which must be 1 — and 2³ ÷ 2⁴ is 2⁻¹, which must be a half. Neither is a convention; both are forced by the rule that already existed.
The fractional case follows from the third. If x^(1/2) is raised to the power 2 the exponents multiply to give x¹, so x^(1/2) squared is x — which is the definition of a square root.
Which roots are worth recognising?
The perfect squares, on sight: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Knowing them makes estimating everything else fast, because any number sits between two of them.
√150 is between 12 and 13 and much closer to 12, because 144 is only 6 away and 169 is 19 away. That is usually enough for a sanity check, and it is the skill that catches an answer that is out by a factor of ten.
Odd roots of negative numbers are real and even roots are not. The cube root of −27 is −3, because −3 × −3 × −3 is −27; the square root of −27 has no real value, because no real number squares to a negative.
Why does a ratio simplify the same way a fraction does?
Because it is one. A ratio simplifies by dividing both sides by their greatest common divisor — 6 : 8 becomes 3 : 4 — which is exactly the reduction the LCM and GCD article describes for fractions.
The three things ratios are used for cover almost everything. Simplifying puts a measured relationship into its clearest form. Solving a proportion — A : B = C : x — multiplies B by C and divides by A. Splitting a quantity by a ratio divides it into the total number of parts and hands out the right number to each.
That last one is the one people get wrong. Splitting 240 in the ratio 3 : 5 means eight parts of 30, giving 90 and 150 — not dividing by 3 and by 5.
Scientific notation is the same machinery doing a different job. Writing 6.022 × 10²³ separates the significant digits from the scale, which is why it is the standard form anywhere numbers span many orders of magnitude — and why a calculator switches to it rather than printing twenty-three zeros.
It also makes multiplication trivial: multiply the mantissas and add the exponents, which is the first rule above applied to base ten. That is the whole reason logarithms were invented, and why slide rules worked.
Is the golden ratio special or mythologised?
Both, and it is worth separating them. φ is (1 + √5) ÷ 2, about 1.6180339887, and its defining property is genuinely elegant: removing a square from a golden rectangle leaves a smaller golden rectangle, indefinitely.
It is also the limit of the ratio between consecutive Fibonacci numbers, which is why 5 and 8, 8 and 13, 13 and 21 all approximate it — and why the kilometre-to-mile trick works, since 1.618 is within half a per cent of 1.609.
The design mythology is much weaker than the mathematics. Claims that the Parthenon, the Great Pyramid and assorted Renaissance paintings were built to φ generally rely on choosing which edges to measure, and they do not survive being checked against the actual dimensions.
Questions people ask
Why is anything to the power zero equal to one? Because dividing a power by itself gives both x⁰ and 1. It is forced by the division rule rather than defined separately.
What is 0⁰? Contested, and defined as 1 in most combinatorial contexts because that is what makes the formulas work. Analysis sometimes leaves it undefined.
How do I estimate a cube root? Between the nearest perfect cubes — 1, 8, 27, 64, 125, 216. The same bracketing method as square roots, with a shorter list to remember.
Does a ratio have to be whole numbers? No, and reducing it to whole numbers is usually clearer. A ratio of 1 : 1.618 and one of 1000 : 1618 describe the same relationship.
Why does a calculator refuse the square root of a negative? Because it is working in real numbers. The answer exists in the complex numbers as an imaginary value, which is a different mode rather than a different button.
Roots are powers, negative exponents are reciprocals, and three rules cover the rest. The square root calculator handles any nth root, the exponent calculator covers the negative and fractional cases, the ratio calculator does all three ratio modes, and the golden ratio calculator is there for the one ratio with a mythology attached.