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!

Solution 1:

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 $a$).

Solution 2:

$$10^{3.5}=10^3\cdot 10^{.5}=10^3 \cdot 10^{\frac{1}{2}}=10^3 \sqrt{10}$$

Half power doesn't mean half of the number, it means square root.

Solution 3:

$10^{3.5}$ 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 $10^{3.5} = 3162.27...$.