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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!

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    $\begingroup$ write 3.5 as $\frac {7}{2}$ so 7 is normal power and 1/2 is square root $\endgroup$ May 11, 2013 at 22:17
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    $\begingroup$ I say leave this question open. This question is actually excellent. The OP has given a good well defined question and I believe that a good answer might actually help the OP with his/her understanding of the topic. $\endgroup$
    – Thomas
    May 12, 2013 at 18:58
  • $\begingroup$ I just want to point out the flaw of logic in your argument. What did you expect $1^{0.5}$ to be ? $\endgroup$ May 12, 2013 at 19:50

6 Answers 6

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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 $a$).

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  • $\begingroup$ How about irrational numbers? Such as a^(1/9) ? $\endgroup$
    – Rafi Kamal
    May 12, 2013 at 8:43
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    $\begingroup$ @Rafi: Where's the irrational number? $\endgroup$ May 12, 2013 at 8:45
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    $\begingroup$ @RafiKamal That requires much more work. You can see this answer for a quick definition. However $1/9$ is not an irrational number. $\endgroup$
    – egreg
    May 12, 2013 at 8:46
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$$10^{3.5}=10^3\cdot 10^{.5}=10^3 \cdot 10^{\frac{1}{2}}=10^3 \sqrt{10}$$

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

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  • $\begingroup$ While very true, it doesn't help with the visualization part of the question: if $10^3 = 10\cdot 10 \cdot 10$, then $10^{3.5} =\ 10\cdot 10 \cdot 10 \cdot\ ?$ $\endgroup$ May 12, 2013 at 1:09
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    $\begingroup$ @Virtlink how about $10^3 = 10^1 * 10^1 * 10^1, so 10^3.5 = 10^1 * 10^1 * 10^1 * 10^0.5? (sorry, can't figure out how to format that with tex) $\endgroup$ May 12, 2013 at 1:45
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    $\begingroup$ @Virtlink $10^{3.5}=10\cdot 10 \cdot 10 \cdot \sqrt{10}$. $\endgroup$
    – N. S.
    May 12, 2013 at 2:50
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$10^{3.5}$ 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 $10^{3.5} = 3162.27...$.

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  • $\begingroup$ Thank you so much, so many good answers!!, no i havent learned about square roots yet, but im on it now. $\endgroup$
    – BjarkeCK
    May 11, 2013 at 22:32
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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 $3163.3$.

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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.

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

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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.

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