# 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 | 4 | 16 | 64 |

Do you notice anything about the numbers?

Time t in hours | 0 | 1 | 2 | 3 |
---|---|---|---|---|

Number x of bacteria | 4^{0}=1 | 4^{1}=4 | 4^{2}=16 | 4^{3}=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 $$ (a ^ m) ^ n = a ^ (m * n) $$.

- What is the attraction of visiting Mars
- Can I load old films from the 1940s
- What are the real assets of a nation?
- Why not keep SMS timestamps
- Which is the best hoverboard in India
- What is the best alternative to LawPay
- How will Kolkata Knight Riders win
- What is meant by NSS
- We should turn our attention to fraudsters
- How to pronounce Parker in Spanish
- What ended your relationship with your spouse
- Why does society demonize pedophiles
- Why didn't Canada or Russia annex Alaska?
- What makes science philosophical
- Can you hack someone for me
- How is SVNIT Surat for mechanical engineering
- Which metal is used for the heating element
- What's your most embarrassing bathroom experience
- How do I feel loved
- What do you think a chatbot does
- Does Kamala Harris support single-paying health care?
- Why am I talking to myself 1
- Radioactive decay always produces light
- Is the Palmer report valid?