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One thing that always confuses me about exponent is that how to visualize it. Suppose we have

$$2^3$$ This means that we multiply 2 three times $2^3 = 2\times2\times2$

But what if the exponent is a decimal such as $2^{2.345}$ Does this mean that we must multiply 2 2.345 times ? I'm confused on how to represent this

Kain
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  • You can ask the same question about multiplication. $2\times 3=2+2+2$. What about $2\times\pi$? The idea is that we continuously extend our intuition about these operations on naturals to larger spaces in sensible ways. – Rushabh Mehta Jul 30 '21 at 11:59

1 Answers1

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Suppose that we have a number $x$ such that $x^{2} = a$ where $a$ is something finite. Then, we can let $x = a^{\large\frac{1}{2}}$. Now, assume that we have a number $x$ such that $x^{n} = a$ where $n \in \mathbb{N}$. We can let $$x = a^{\large\frac{1}{n}}.$$

Now, we can raise $x$ to an arbitrary power $m$ such that $$x^{m} = a^{\large\frac{m}{n}}.$$

In your case, the exponent is $2.345$ and is equal to $\frac{469}{200}$. Hence, $$2^{2.345} = \sqrt[\large200]{2^{469}}.$$

soupless
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