irrational number

in mathematics, the irrational numbers are all the real numbers that are not rational numbers
that is, irrational numbers cannot be expressed as the ratio of two integers

some examples of irrational numbers are $\pi$ and $\sqrt{2}$

the irrational number that use the root operator can actually be expressed as exponent form

irrational numbers can be treated as variables in their own right, for example $\pi$ and $e$ - we treat them as variables
but in real world applications it is possible to treat them as decimals for rough estimations

one might prefer to see a number such as $\sqrt{2}$ as $(2{)}^{0.5}\vee 2^{\frac{1}{2}}$ to make the calculation steps clearer

like variables, irrational numbers can have coefficients. $\sqrt{2}$ is an irrational number, and when you double it to get $2\sqrt{2}$ it is still an irrational number.

irrational numbers can be simplified with the use of perfect squares and factoring.
$\sqrt{4}=2$
$\sqrt{8}=\sqrt{4\cdot 2}=2\sqrt{2}$
$\sqrt{32}=\sqrt{16\cdot 2}=4\sqrt{2}$

likewise, they can be unsimplified when you convert the coefficient into it
$2\sqrt{2}=\sqrt{4\cdot 2}=\sqrt{8}$
$4\sqrt{2}=\sqrt{16\cdot 2}=\sqrt{32}$

as $a^{\frac{1}{b}}=\sqrt[b]{a}$, operations involving irrational numbers can be done by following the exponent rules/

just like variables, like terms can be collected.
$\sqrt{7}+\sqrt{7}=2\sqrt{7}$

unlike terms cannot be collected.
$\sqrt{2}+\sqrt{7}=\sqrt{2}+\sqrt{7}$

when there are many different irrationals, make sure that you get the calculations right
$\sqrt{2}+\sqrt{7}-\sqrt{5}+\sqrt{2}-4\sqrt{7}=2\sqrt{2}-\sqrt{5}-3\sqrt{7}$

like terms may combine

$\sqrt{7}\cdot \sqrt{7}=7$
$(7{)}^{0.5}\cdot (7{)}^{0.5}=(7^{0.5}{)}^2=7^{0.5\cdot 2}=7^1=7$

coefficients may combine
$2\sqrt{7}\cdot 7\sqrt{7}=2\cdot 7\cdot 7=98$

unlike terms may be rearranged
$2\sqrt{12}\cdot \sqrt{48}=2\sqrt{4\cdot 3}\cdot \sqrt{4\cdot 4\cdot 3}=4\sqrt{3}\cdot 4\sqrt{3}=4\cdot 4\cdot 3=48$