How do you calculate with fractional exponents

New exponents

$$ 2 ^ 3 $$, $$ (- 25) ^ 2 $$, $$ x ^ -2 $$, $$ (1/4) ^ 2 $$, $$ 1,5 ^ -1 $$

You are familiar with these powers: different numbers as a base and positive and negative whole numbers as an exponent.

But: The exponents can too Fractions be like in $$ 2 ^ (1/2) $$!

Huh? $$ 2 ^ 3 = 2 * 2 * 2 $$, but how can that work with a fraction ...

That is determined by the root! Here we go:

Fractions $$ m / n $$ as an exponent

The exponent can also be another fraction. Look at the term $$ x ^ (6/7) $$.
How is that supposed to work now?

$$ x ^ (6/7) $$
is the same as:
$$ x ^ (6 * 1/7) $$

Power laws:
$$ (x ^ 6) ^ (1/7) $$

$$ n $$ - take the root for $$ n = 7 $$:
$$ root 7 (x ^ 6) $$

So: $$ x ^ (6/7) = root 7 (x ^ 6) $$

For a number a: $$ a ^ (m / n) = root n (a ^ m) $$
Here a is a real number greater than 0, n is a natural number greater than 1 and m is an integer.
$$ a in RR $$ and $$ a> 0 $$; $$ n in NN $$ and $$ n> 1 $$; $$ m in ZZ $$.


Most of the time you calculate these powers or roots with a pocket calculator. With some calculators you shouldn't forget the brackets:
[Image of the input: x ^ (6/7)]





And this is how it works in general:
$$ x ^ (a / b) $$

$$ x ^ (a * 1 / b) $$

$$ (x ^ a) ^ (1 / b) $$

$$ root b (x ^ a) $$

And in practice?

Powers with rational exponents occur in bacterial growth.

One type of bacteria multiplies in such a way that its number quadruples after an hour.

Time t in hours 0 1 2 3
Number x of bacteria 1 416 64

Do you notice anything about the numbers?

Time t in hours 0 1 2 3
Number x of bacteria 40=1 41=4 42=16 43=64


You can write that in a formula: $$ \ text {number of bacteria} = 4 ^ (\ text {number of hours}) $$ or $$ x = 4 ^ t $$ for short.

With the formula you can count the number of bacteria after a half Calculate hour. Now the roots come into play.

$$ x = 4 ^ (1/2) = sqrt (4) = 2 $$

Or after $$ 2.5 $$ hours?

$$ x = 4 ^ (2.5) = 4 ^ (5/2) = 4 ^ (5 * (1/2)) = (4 ^ 5) ^ (1/2) = sqrt (4 ^ 5) = sqrt (1024) = 32 $$

After 2.5 hours there were 32 bacteria.

For this calculation you already needed a few rules from fractions and power laws like $$ (a ^ m) ^ n = a ^ (m * n) $$.