Multiplication, What is It? What is multiplication? Upon review logarithms, and square roots, I realized that I have no intuitive grasp of multiplication-well no more so than I have for addition. Is it simply another thing we need to memorize? I understand that things like $\sqrt2$ could just be memorize as the thing, that when applied to itself, gives two... But this makes a lot less sense to me then thinking about things like $2+2=4$. Logarithms are a function that express a number as a power of some base, but when you get fractions for the logarithms, I don't really know what that means-What is $2^{1.2}$, or any decimal power for that matter.
Upon rereading the link posted below, I think I can repose my question a bit. If these operations are axiomatic, as I felt they were, the idea of a square root is the inverse of exponentiation. How do I grasp exponentiation intuitively? Multiplication and addition have natural roots in our minds, if exponentiation is also fundamental, what is it? Also when is exponentiation not just repeated multiplication? 
 A: First, you shouldn't find expressions like $\sqrt{2}$ weird.  Saying that $\sqrt[a]{b}$, where $a$ and $b$ are positive integers, is the number whose $a^\text{th}$ power is $b$ is perfectly fine.  Just think about the graph of $x^a$.  This is fudging a little bit, but we know $x^a$ is continuous, meaning it's a solid line, and we know it goes up to $\infty$ in the positive direction.  So we know that the lines $y = x^a$ and $y = b$ intersect at some point.  When you write down $\sqrt[a]{b}$ that's just notation for the $x$-coordinate of that point of intersection.
Now if you're fine with $\sqrt[a]{b}$ then you should be fine with raising a number to a fraction.  When we write $x^\frac{a}{b}$ we just mean $\sqrt[b]{(x^a)}$ or $(\sqrt[b]{x})^a$ (they're the same).
After you're fine with raising numbers to fractions you might ask what does $x^a$ mean if $a$ is $\pi$ or some other number that's not a fraction.  This is maybe farther than you want to go at the moment, so I won't give you a detailed answer (I'm sure this questions exists elsewhere on the site).  I'll just say that $x^a$ is the number that $x^b$ gets closer to as you choose fractions $b$ that get closer and closer to $a$.
A: The answer you learns at your mother's knee, that it's "repeated addition", is part of the truth.  That is one manifestation of multiplication, and should probably be regarded as the most important one.
You should NEVER consider anything in mathematics just another thing you need to memorize.
$3\times4$ is the sum of three $4$s; it is $4+4+4$.
$4\times3$ is the sum of four $3$s; it is $3+3+3+3$.
There is also this item.
You can do with real numbers what the item above does with complex numbers.  Here we see $4\cdot3$.
$$
\begin{array}{cccc}
\hline \\
0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
0 &   &   & 1 &   &   & 2 &   &   & 3 &    &    & 4  
\end{array}
$$
Here we see $4\cdot(-3)$:
$$
\begin{array}{cccc}
\hline \\
-12 & -11 & -10 & -9 & -8 & -7 & -6 & -5 & -4 & -3 & -2 & -1 & 0 \\
\phantom{-}4 & & & \phantom{-}3 & & & \phantom{-}2 & & &\phantom{-} 1 & & & 0
\end{array}
$$
Here we see $(-2)\cdot(-3)$ and $2\cdot(-3)$:
$$
\begin{array}{cccc}
\hline \\
-6 & -5 & -4 & -3 & -2 & -1 & 0 & \phantom{-}1 & \phantom{-}2 & \phantom{-}3 & \phantom{-}4 & \phantom{-}5 & \phantom{-}6 \\
\phantom{-}2 & & &\phantom{-} 1 & & & 0 & & & -1 & & & -2
\end{array}
$$
In the same way we could look at $(-1.5)\cdot(-3)$, etc.
A: Exponents are a way of showing multiplication when what is being multiplied is being multiplied by itself.  Saying $4^{1.2}$ is just saying that you are multiplying $4$ by itself $1.2$ times.  
