10 to the power of 3.5: $10^{3.5}$ So $10^3 = 10\times 10\times 10 = 1000$, this is really easy to understand.
But what about: $\,10^{3.5}\,?\,$ My logic would suggest this was 

$10\times 10 \times 10\times 5 = 5000,\;$ but the calculator says it's 3162.27...

Can someone illustrate how the calculator calculates the power of when the number is with decimals? 
Please keep in mind that I'm a newbie to mathematics!
 A: Your mistake is computing $10^\frac{6}{2} \cdot \frac{10}{2}$ instead of $10^\frac{6}{2} \cdot 10^\frac{1}{2}$. Although 5 is halfway to ten when working with addition, it is past halfway to ten when working with multiplication.
$10^{3.5}$ is equivalent to $10^\frac{7}{2}$. You can perform the '7' and 'over 2' parts of the exponentiation separately. That is to say: $10^\frac{7}{2} = (10^7)^\frac{1}{2}$. That's just $\sqrt{10^7}$, which can be reduced to $10^3 \sqrt{10}$, which is approximately $3163.3$.
A: With $10^{0.5}$ you have to do a "half multiplication", not a multiplication by half. What this "half multiplication" is doesn't follow from any universal law, but only by extending in a coherent way the rules valid for integer exponents.
Since, for integer $m$ and $n$ you have
$$
a^{m+n}=a^m\cdot a^n
$$
you can derive also that
$$
(a^m)^n=a^{mn}
$$
(just repeat the multiplications and count the factors). So, what $a^{3.5}$ should mean? Well, a possible choice comes from doing
$$
a^7=a^{3.5\cdot 2}\overset{*}{=}(a^{3.5})^2
$$
where the equals sign marked with $*$ is where we apply an extension to the  rule above.
Thus one can try defining
$$
a^{3.5}=\sqrt{a^7}.
$$
This is how actually exponentiation to a rational is defined:
$$
a^{\frac{p}{q}}=\sqrt[q]{a^p}
$$
and it can be shown that the rules
$$
a^{x+y}=a^x\cdot a^y,\qquad
(a^x)^y=a^{xy}
$$
continue to hold for all rational numbers $x$ and $y$ (and positive $a$).
A: N.S. is exactly correct; $10^.5 = \sqrt{10}$. 
For a small amount of reasoning behind it.
By exponent laws, we know $10^x \times 10^y = 10^{x+y}$. $10^.5 \times 10^.5 = 10^1 = 10$. Since we want to figure out what $10^.5$ is, let's sub in $x$ for it.
$x \times x = 10 \rightarrow \sqrt{x^2}=\sqrt{10} \to x = \sqrt{10}$
A: $$10^{3.5}=10^3\cdot 10^{.5}=10^3 \cdot 10^{\frac{1}{2}}=10^3 \sqrt{10}$$
Half power doesn't mean half of the number, it means square root.
A: $10^{3.5}$ is equal to $10*10*10*10^{0.5}$. So you just need to know what $10^{0.5}$ is.
One of the property of exponent is this.
$(10^x)^y = 10^{xy}$, i.e. the power of a power, is just the exponents multiplied.
So if $(10^{0.5})^2=x^2=10^1$ when $x=10^{0.5}$ then we just solve for $x^2 = 10$.
I'm not sure if you learned about square roots, yet, but $x=\sqrt{10}$. 
Since $\sqrt{10}$ is approximately equal to 3.16227, your calculator gives you $10^{3.5} = 3162.27...$.
A: There was a question about irrational numbers. For example, how would
$10^{\pi}$ be calculated?
The trick is to use $ln(x)$ and $e^{x}$ which are inverse functions, that is,
$e^{ln(x)} = x$
If you then use $x = a^{n}$, you'll get
$a^{n} = e^{ln(a^{n})} = e^{n  ln(a)}$
or
$10^{\pi} =  e^{\pi ln(10)}$
Both $ln(x)$ and $e^{x}$ can be easily calculated.
