Laws of Indices :
- $a^{m} \times a^{n}= a^{m+n}$
- $\frac{a^{m}}{a^{n}}= a^{m-n}$
- $(a^{m})^{n}=a^{mn}$
- $(ab)^{n} = a^{n}b^{n}$
- $\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}$
- $a^{0}=1$
Surds:
When we can't simplify a number to remove a square root (or cube root etc) then it is called a surd.
Rational Number:
A real number that can be written as a simple fraction then it is called rational number
e.g. 1.5 = $\frac{3}{2}$
Irrational Number:
A real number that can't be written as a simple fraction is called Irrational number.
e.g. $\pi$ = 3.1415926535897932384626433832795
Let a be rational number and n be a positive integer such that $a^{\frac{1}{n}}=\sqrt[n]{a}$ is irrational. Then $\sqrt[n]{a}$ is called surd of order n.
Laws of Surds
- $\sqrt[n]{a}=a^{\frac{1}{n}}$
- $\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$
- $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$
- $(\sqrt[n]{a})^{n}= a$
- $\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$
- $(\sqrt[n]{a})^{m} = \sqrt[n]{a^{m}}$
