I need "intuition" about fraction exponents, like $4^{1.2}$. What exactly is it meant to do with number $4$? [duplicate]

$4^3$ is $4\cdot 4\cdot 4$

Then $4^{1.2}$ is what, $4\cdot\dotso$??

What happens to the number with a fraction exponent when we try to represent it only using numbers and basic operations (like $+$, $-$, $\div$, and $\cdot$)?

$4^{1.2} = 4*4^{0.2}$; but there's still a fractional exponent, $0.2$, there. I can't represent $4^{1.2}$ only using basic operations.

Is there a way to do that? Is it impossible fundamentally in math, or is it possible using imaginary numbers?

i checked answers form this 8 year old question but it's not giving the answer i'm looking for.

Solution 1:

Before trying to answer the question you asked let's look at another.

Clearly you know what $$ 4 \times 3 $$ means. After all; multiplication is just repeated addition, so $$ 4 \times 3 = 4 + 4 + 4 = 12. $$ But what in the world is $$ 4 \times 1.2 ? $$ You can't add $4$ to itself $1.2$ times. You have to extend the definition of multiplication to allow for multiplying by a fraction like $1.2 = 6/5$. That's just what you learned about in elementary school when you struggled with fractions. Understanding exactly what $4 \times \sqrt{2}$ or $4 \times \pi$ mean is even subtler. You probably just did this with decimal approximations.

Now to your question. In order to understand $4^{1.2}$ you have to extend the definition of exponentiation. You can't just think of it as repeated multiplication. Doing that is subtle. It starts with thinking about the one half power and seeing why you want $9^{1/2}$ to be $3 = \sqrt{9}$. That's what the other answers are trying to tell you. There is no way to do it with the kind of expression you ask for in a comment.

(whole number)^(optional - whole number)/(some whole number)^(optional - whole number)

The best I can do for you is to think about $$ 4^{1.2} = 4 \times 4^{1/5}. $$ Now $4^{1/5}$ is some number. Call it $z$. Clearly $z$ is not just "$20\%$ of $4$". Whatever value $z$ is must satisfy $$ z^5 = (4^{1/5})^5 = 4 $$ so $z$ is the fifth root of $4$, or $\sqrt[5]{4}$. $$ $$

Solution 2:

I'm going to assume you're only referring to rational powers here. Not sure if this is what you're looking for, but I think of it as an inverse operation to give you something resembling the repeated multiplication form you are looking for.

For instance, if you're trying to find the square root of $4$, you're trying to solve the following equation:

$$x=4^{0.5};$$

so instead of $4\times 4\times\cdots$ for some number of times, you're looking at $x\times x=4$. For any rational power $r=p/q$, you're solving two things:

• The first is the $p$-th power, or $4\times 4\times \cdots$ for $p$ times; say this value is $k$.
• Then $x\times x \times \cdots$ for $q$ times and equate that with $k$.