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Your graphing calculator can compute x\sqrt{x} easily, but computer algorithms almost always rewrite that root as x0.5x^{0.5} behind the scenes.
Writing roots as fractions lets you apply every standard exponent rule without ever drawing a radical sign.
How do we bridge these two mathematical worlds seamlessly?

Roots as Fractions

Every radical expression contains an index (the small number indicating the degree of the root) and a radicand (the quantity sitting underneath the radical symbol).
A rational exponent is simply an exponent written as a fraction, where the numerator represents the power and the denominator represents the root.
Why are we allowed to turn a root into a fraction in the first place?

Why Rational Exponents Work

In the 1650s, English mathematician John Wallis formalized fractional exponents so that the power-of-a-power rule, (ap)q=ap⋅q(a^p)^q = a^{p \cdot q}, would work for all numbers, not just integers.
By definition, the square root x\sqrt{x} is the number you square to get xx. If we write x\sqrt{x} as xkx^k, then (xk)2=x1(x^k)^2 = x^1, which means 2k=12k = 1, so kk must equal 12\frac{1}{2}.
Following that exact logic for any positive integer nn, (an)n=a1(\sqrt[n]{a})^n = a^1, so (a1/n)n=an/n=a1(a^{1/n})^n = a^{n/n} = a^1, proving that an=a1/n\sqrt[n]{a} = a^{1/n}.
Now that we know why the rule holds, how do we convert algebraic radicals into powers?