# 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 Number x of bacteria 0 1 2 3 1 4 16 64

Do you notice anything about the numbers?

 Time t in hours Number x of bacteria 0 1 2 3 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) \$\$.