31.09 - Powers and roots

Powers

A power is a number that repeatedly gets multiplied by itself. For example: $2^3=2\times 2\times 2=8$.

• $2^3$ is called a Power
• $2$ is called the Base
• $3$ the Exponent

So in simple terms: a power is a number(base) that multiplies itself the number of times set by the exponent.

Powers with fractions

Info

For more info on fractions, look here.

Fractions are really simple: $(\frac{2}{5}{)}^3=\frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}=\frac{2^3}{5^3}=\frac{2\times 2\times 2}{5\times 5\times 5}$

Powers with a negative base

When the base is negative and between parentheses, the solution is also negative when the exponent is uneven. This also works vice versa; the solution is positive when the exponent is even. Keep in mind that this only counts when the base is between parentheses, otherwise the solution is also negative.

$$\begin{array}{rl}-5^3& =-125\\ (-5{)}^3& =-125\\ -5^4& =-625\\ (-5{)}^4& =625\end{array}$$

The explenation is this:

$$\begin{array}{rl}\text{a is an integer}& \\ \text{n is a natural number}& \\ \boxed{\begin{array}{rl}{a}^1& =a\\ a^{n+1}& =a\times a^n\end{array}}& \\ & \\ (-5{)}^4=(-5)(-5{)}^3...& =(-5)(-5)(-5)(-5)=\\ (-1\times 5)(-1\times 5)(-1\times 5)(-1\times 5)& =\\ (-1)(-1)(-1)(-1)\times 5^4& =\\ (-1)(-1)(-1)(-1)\times 5^4& =-1\times 5^4\end{array}$$

As you can see, the result when using parentheses is negative.
Now let’s do the same but without the parentheses:

$$\begin{array}{rl}\text{a is an integer}& \\ \text{n is a natural number}& \\ \boxed{\begin{array}{rl}{a}^1& =a\\ a^{n+1}& =a\times a^n\end{array}}& \\ & \\ -5^4=-5\times 5^3...& =-5\times 5\times 5\times 5=\\ (-1\times 5)\times 5\times 5\times 5& =\\ (-1\times 5)\times 5^3& =-1\times 5^4\end{array}$$

Powers with a negative exponent

Negative exponents may seem like its the same as a negative base, but that is not true actually. For exponents is true that $a^n÷a=a^n-1$: dividing a power by its base lowers the exponent by 1.

Now take this example:

$$\begin{array}{rl}{a}^n÷a& =a^n-1\\ a^1& =a\\ a^1÷a& =a^0\\ a^0÷a& =a^{-1}\\ 5^1÷5=5÷5& =1=5^0\\ 5^0÷5=5^{-1}& =1/5=0.2\\ 5^{-1}÷5& =0.2÷5=0.04\end{array}$$

As you can see, as soon as the exponent turns negative, the base will be divided by itself the number of times set by the exponent instead.

Products, quotients and powers of powers with equal bases

A rule of thumb

Since exponents work on an exponential scale instead of a linear scale, if the exponent go up by one, you actually multiply again. Following this rule, multiplying again is the same as adding one to the exponent. This also works the other way around: If the exponent goes down by one, you divide once time and if you divide once, the exponent goes down by one.

If you multiply two powers with equal bases, you add the exponents together and the base stays the same: $5^7\times 5^3=5^{7+3}=5^{10}$.
If you divide two powers with equal bases, you subtract the exponents from one another and the base stays the same: $5^7÷5^3=5^{7-3}=5^4$.

if you add an exponent to the exponent(power of a power), you multiply the exponent and the base stays the same: $(5^3{)}^7=5^{3\times 7}=5^{21}$.