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

Question

$4^3$ 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.

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

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

Answer 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 $\sqrt[5]{4}$. $$ $$

Answer 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 $k$.

Answer 3

I think graphing the exponential function will help you come to an intuitive understanding of it.

So here's a plot of $y = 4^x$ for $x = [0, 4]$:

But how do we fill in the values in-between these whole number exponents?

Well let's try plotting some other exponential functions. Here's $y = 2^x$:

You might not be able to tell just from looking, but it's the same shape. All of the points on the first plot are there on the second plot. $4^4 = 2^8 = 256$.

We might now guess that $4^{1.5} = 4^{3/2} = 2^3$, and indeed, that's true, both equal 8. We can fill in the plot for 0.5, 1.5, 2.5 and 3.5:

Unfortunately it's not as simple as $a^{b/c} = (a/c)^b$, instead the rule works out to be

$$a^{b/c} = \sqrt[c]{a^b}$$

So it's a little more complicated to work out, but the exponential function is continuous, and all the intermediate values can be calculated. And very occasionally, they work out to be integers too. But I hope this has given you a bit of intuition into how fractional exponents work.

Answer 4

Seems intuitive that $p:=4^{1.2}$ should lie somewhere between $4^1=4$ and $4^2=16$. But where ?

To answer this in a consistent way only relying on integer powers, we make three remarks:

• for any number $q$ and naturals $m,n$, $(q^m)^n=q^{mn}$ is always true;

• it does not seem unreasonable to generalize to $m,n$ not integers;

• accepting this rule, $p^5=(4^{1.2})^5=4^6=4096.$

Now we have to solve the equation

$$p^5=4096,$$ which we can do by trial and error.

For instance, we can write

$$4^5=1024,5^5=3125,6^5=7776,$$ and $p$ should lie between $5$ and $6$.

Then we refine by introducing decimals,

$$5^5=3125,5.1^5=3450.25251,5.2^5=3802.04032,5.3^5=4181.95493$$

and $p$ should lie between $5.2$ and $5.3$.

Continuing this search, we obtain (after very very lengthy computation)

$$p=5.2780316430915770374960078849186\cdots$$

It is possible to show that this number has infinitely many decimals, which never repeat periodically. It is said to be irrational. By means of this trick, we can in principle raise any number to a power that is a fraction. We can also raise to an irrational power by using a close rational approximation of that power.

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In practice, computing arbitrary powers is made easy by means of the so-called logarithms and antilogarithms, using the formula

$$a^b=\text{antilog}(\log(a)\cdot b)$$

where $\log$ and $\text{antilog}$ denote two special functions that can be computed efficiently or tabulated. Observe that taking a power is now replaced by two function evaluations and a product.

(Historically, logarithms were introduced to ease the computation of products by hand, using $$a\cdot b=\text{antilog}(\log(a)+\log(b))$$ which trades a multiplication for an addition.)

Answer 5

• Let $a\geq0,$ and $m,n$ be positive integers.

  Then, by definition, $a^\frac mn$ equals $(a^\frac1n)^m.$

• $a^\frac1n$ refers to the set of the $n$th roots of $a.$

  While every $n$th root is an algebraic number, most are not rational.

• This plot represents $4^{0.2}=4^\frac15$ on the complex plane:

  

  Here, only the rightmost number (orange dot) is real. Its value $\approx1.32$ equals the circle's radius. It is typically denoted by $\sqrt[5]4,$ and is an irrational number, i.e., it cannot be expressed as the ratio of two integers, and as such is not expressible by merely dividing (and/or adding, subtracting, multiplying) whole numbers.