Fractional exponents can all be expressed through radicals. Edit: I realize the actual question was about irrational exponents. I think the most intuitive way to thing about it is to imagine a sequence of rational numbers that converge to the irrational exponent. read more

Fractional exponents let us figure out how much money you have in the in-between times. Let's represent time by [math]t[/math], which is in years. Then the number of doublings is [math]t/12 = x[/math], so your money is [math]M = 2^{t/12}[/math] If we can figure out how much money you have at a given time, we've figured out fractional exponents. read more

If we fix $a>0$, $f(x)=a^x$ is continuous on $\mathbb R$. The intuition behind rational exponents is pretty clear, and one extends from the rationals to all reals in this manner. read more

Source: math.stackexchange.com

Source: slideshare.net

What Is Intuition, And How Do We Use It?

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