"Multiply 2 by itself three times" makes sense: 2 \times 2 \times 2 = 8. But "multiply 2 by itself one-half times"? What does that even mean? You can't multiply something a fractional number of times any more than you can have half a goat. And yet 2^{1/2} is a real number that shows up on every calculator, equal to \sqrt{2} \approx 1.414. Where does the square root come from?

The answer is: the "count the multiplications" picture was a motivating definition, not the final one. Once you start extending the rule to fractions, you have to give up the counting picture and let the laws of exponents do the defining instead.

The counting picture breaks — admit it up front

For positive integers, a^n means a \times a \times \dots \times a with n factors. This works beautifully for a^2, a^3, a^4. It already got a bit strange for a^1 ("multiply by a exactly once" = just a) and for a^0 ("multiply by a zero times" = a little metaphysical, but 1).

For a^{1/2}, the counting picture is nonsense. You cannot halve a multiplication. So either a^{1/2} has no meaning, or the definition of "exponent" has to be widened.

Mathematics chooses: widen the definition. And the widening is forced by consistency with the laws of exponents that already work for integer powers.

The one-line derivation

The key law is power of a power: (a^m)^n = a^{m \cdot n}.

Plug in m = \tfrac{1}{2} and n = 2:

\left(a^{1/2}\right)^2 \;=\; a^{(1/2) \cdot 2} \;=\; a^1 \;=\; a

So a^{1/2} is a number whose square is a. By definition, that number is \sqrt{a}:

\boxed{a^{1/2} \;=\; \sqrt{a}}

There is no other choice. If you defined a^{1/2} to be anything else, the power-of-a-power law would break. The square-root interpretation is forced by requiring the laws of exponents to continue working.

Concrete values

Let's walk through the picture for 2^{1/2}:

And the last line is the sanity check: if 2^{1/2} means something that, when multiplied by itself, gives 2, it has to be \sqrt{2}. You can verify numerically: 1.414 \times 1.414 \approx 1.999 \approx 2.

Similarly:

The general rule, same argument with n in place of 2:

a^{1/n} \;=\; \sqrt[n]{a}

The exponent \tfrac{1}{n} corresponds to the n-th root. This is the entire content of Roots and Radicals — roots are fractional exponents in disguise.

The continuous curve — where fractional exponents live

The curve two to the x with landmark fractional exponents labelledA smooth rising curve of y equals two to the x. Five integer landmarks are marked: x equals zero gives one, x equals one gives two, x equals two gives four, x equals three gives eight. Between them, fractional values are marked. At x equals zero point five the curve passes through the square root of two, about one point four one four. At x equals one point five the curve passes through two times the square root of two, about two point eight two eight. At x equals two point five the curve passes through four times the square root of two, about five point six five seven. The curve is continuous — no gaps, no jumps — which is why every real exponent maps to a real positive value. 0 1 2 3 4 (x) 1 2 4 8 2⁰ = 1 2^(1/2) = √2 ≈ 1.414 2¹ = 2 2^(3/2) = 2√2 ≈ 2.828 2² = 4 2^(5/2) = 4√2 ≈ 5.657 2³ = 8
The curve $y = 2^x$ is continuous. The black dots are the familiar integer powers ($2^0 = 1$, $2^1 = 2$, $2^2 = 4$, $2^3 = 8$). The coloured dots are the fractional exponents, sitting halfway between consecutive integers on the $x$-axis. At $x = \tfrac{1}{2}$, the curve passes through $\sqrt{2} \approx 1.414$; at $x = \tfrac{3}{2}$, it passes through $2\sqrt{2} \approx 2.828$. Every real exponent lives somewhere on this smooth curve.

Here is a useful way to think about it. The integer powers 2^0, 2^1, 2^2, 2^3, \dots give you isolated points at (0,1), (1,2), (2,4), (3,8), \dots on the graph. Connect those points with a smooth curve, and you have the function 2^x for all real x, including the fractional values. The fractional exponents are just the y-coordinates of the in-between points you would hit if you walked along the curve between the integer landmarks.

The curve has no gaps. At x = \tfrac{1}{2} it passes through \sqrt{2} \approx 1.414 — a value halfway along the curve between the 2^0 = 1 point and the 2^1 = 2 point (in logarithmic distance, which is the right measure here).

A physical reading — doubling time

For a physical picture, imagine a population that doubles every hour. After t hours, it is P(t) = P_0 \cdot 2^t of its original size. You are asking: what is the population after half an hour?

The population at half an hour is not half the increase (that would be 1.5 P_0, the linear guess). It is the geometric mean of the start and end values: \sqrt{P_0 \cdot 2P_0} = \sqrt{2} P_0. Fractional exponents naturally give the geometric middle, which is the right notion of "halfway" for a process that multiplies rather than adds.

Exponents of the form \tfrac{m}{n} continue this: 2^{m/n} is the population after \tfrac{m}{n} doublings, equivalently \sqrt[n]{2^m}.

General fractional exponent

Combine the pieces. a^{m/n} can be read two equivalent ways:

a^{m/n} \;=\; \left(a^{1/n}\right)^m \;=\; \left(\sqrt[n]{a}\right)^m

or, by commuting inside the exponent-of-an-exponent,

a^{m/n} \;=\; \left(a^m\right)^{1/n} \;=\; \sqrt[n]{a^m}

Both give the same answer. 8^{2/3} can be computed as (8^{1/3})^2 = 2^2 = 4, or as (8^2)^{1/3} = 64^{1/3} = 4. Pick whichever is easier arithmetically.

So what does "raised to a fractional power" really mean?

It is not "multiply a fractional number of times." That picture was always a crutch for integer exponents. The real definition is:

a^x is the unique continuous function of x that agrees with repeated multiplication at integer points, and obeys all the laws of exponents everywhere else.

For rational x = \tfrac{m}{n}, the value is \sqrt[n]{a^m}. For irrational x (like a^{\sqrt{2}}), it is the limit of a^r as rational r \to x — a construction that only makes full sense once you reach the Limits Introduction chapter.

The takeaway: your intuition of "counting multiplications" was never the final definition. It was a scaffold. The laws of exponents are the real definition, and they extend the idea of a^x cleanly to every real x. A fractional exponent is a root in disguise. An irrational exponent is a limit of rational exponents. Every one of them lives on the same smooth curve y = a^x.

Related: Exponents and Powers · Roots and Radicals · Exponent Slider: Watch 2^x Sweep Through 1/8, 1/4, 1/2, 1, 2, 4, 8