10 to the power of 3.5: $10^{3.5}$

Question

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!

Answer 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 $a$).

Answer 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.

Answer 3

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.

$x \times x = 10 \rightarrow \sqrt{x^2}=\sqrt{10} \to x = \sqrt{10}$

Answer 4

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

Answer 5

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 $3163.3$.

Answer 6

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.