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.

The power $2^3$ is also called two to the third.

Powers with fractions

Info

For more info on fractions, look here.

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

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 actually not true. 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.

Since these numbers get small really quick, its easier to note it down as a fraction.

The denominator of a power with a negative exponent is the power with the same, but positive, exponent.

$$\begin{array}{rl}{7}^{-4}& =?\\ 7^4& =2401\\ 7^{-4}& =\frac{1}{2401}\end{array}$$

The same rules apply for when the base is negative. As mentioned in fractions which are negative: it doesn’t matter where the minus is added.

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: $(5^3{)}^7=5^{3\times 7}=5^{21}$.

Roots

Roots are the opposite of powers: solving the root of any number will get you the base of a power with a certain exponent set by the type of root you use.

The radical sign ($\sqrt{x}$)

The symbol of a root is called the radical and is written like the funny vans logo: $\sqrt{x+2}$.

Square roots

Square roots are the opposite of numbers to the second($n^2$): The square root of a number is what number you have to multiply by itself to get that number.

Example: $\sqrt{9}=3$, because $3^2=9$.

Some numbers have infinite numbers as their square root, like 2:

$$\sqrt{2}\approx \mathrm{1.41423562....}$$

Numbers like this will always have an approximation, which is why ≈ is used.

Higher roots

The square root isn’t the only root that exists. For example a cubed root is the opposite of a number to the third($n^3$) and is written as $\sqrt[3]{n}$ (for every positive, integer n). $\sqrt[3]{27}=3$ because $3\times 3\times 3=27$. The same is true for $\sqrt[4]{x}$, $\sqrt[5]{x}$, $\sqrt[6]{x}$…

Standard notation

Standard notation of a root is the most simplified way of writing a root. It is either a whole number(integer) or a root. Standard roots can be used to make calculating the square root easier (sometimes).

For the root to be in standard notation, the following rules have to apply:

1. The number in front of the radical sign is the largest factor possible This means that as many powers have to be eliminated from the number inside the radical sign.
   Keep in mind that there can only be a single root at the end.

   Therefore you shouldn’t simplify (for example) $\sqrt{14}$ to $\sqrt{2}\times \sqrt{7}$, but instead: $\sqrt{12}=\sqrt{2^2\times 3}=\sqrt{2^2}\times \sqrt{3}=2\sqrt{3}$

   Another example:

   $$\begin{array}{r}\sqrt{56}=\sqrt{2^2\times 14}=\sqrt{2^2}\times \sqrt{14}=2\sqrt{14}\end{array}$$

2. There are no fractions inside the radical sign. Instead of having a root of a fraction, we will turn it into a fraction of roots.

   $$\sqrt{\frac{2}{3}}=\frac{\sqrt{2}}{\sqrt{3}}$$

3. There are no roots in the denominator of the fraction. This is done by multiplying the numerator and the denominator by the root from the denominator.

   $$\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{2}\times \sqrt{3}}{\sqrt{3}\times \sqrt{3}}=\frac{\sqrt{2\times 3}}{\sqrt{3\times 3}}=\frac{\sqrt{6}}{3}$$

   Or

   $$\frac{\sqrt[3]{35}}{\sqrt[3]{36}}=\frac{\sqrt[3]{35}\times (\sqrt[3]{36}{)}^{3-1}}{\sqrt[3]{36}\times (\sqrt[3]{36}{)}^{3-1}}=\frac{\sqrt[3]{35}\times (\sqrt[3]{36}{)}^2}{{\sqrt[3]{36}}^4}=\frac{\sqrt[3]{35\times 36^2}}{36}=$$

4. The rational numbers in the expression are simplified

   $$\frac{\sqrt{6}}{3}=\frac{1}{3}\sqrt{6}$$

   Or

   $$\frac{\sqrt[4]{2^9\times 3^6\times 7}}{72}=\frac{2^2\times 3^1\sqrt[4]{2^1\times 3^2\times 7}}{72}=\frac{12\sqrt[4]{126}}{72}=12\times \frac{1}{72}\sqrt[4]{126}=\frac{12}{72}\sqrt[4]{126}$$

   This last part is correct because dividing by 3 or multiplying by $\frac{1}{3}$ is the same.

Rules

Even roots are always positive since there’s no way to multiply a number an even amount of times to get a negative number(two, four, eight negatives are positive). This is why $\sqrt{-}1$ doesn’t exist (for now).

Pile of tips

Writing $\sqrt{4}\times \sqrt{5}$ is the same as writing $\sqrt{4\times 5}$.

$(\sqrt{n}{)}^2=\sqrt{n}\times \sqrt{n}=\sqrt{n^2}=n$. for every positive number of n

Since $\sqrt{n}$ is a divisor of $n$, it means that if $n$ is a divisor of $a$, $\sqrt{n}$ is also a divisor of $a$.

Keep in mind that the number under the radical sign can also be more than a single integer.