Quant · Algebra

Surds and Indices

2 sub-topics · 19 lessons · 60 questions · 12 PYQs

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Indices (Exponents)10 lessons · 33 questions · 7 PYQs

Surds9 lessons · 27 questions · 5 PYQs

Formula sheet

Conjugate product of nested roots
$\sqrt{2+u}\cdot\sqrt{2-u}=\sqrt{4-u^{2}}$
Exponent inequality: base above or below 1
$b^{u}<b^{v}\iff u<v\ (b>1),\quad u>v\ (0<b<1)$
Negative and fractional exponents
$a^{-n}=\frac{1}{a^n},\quad a^{m/n}=\sqrt[n]{a^m}$
Difference of close square roots
$\sqrt{n+d}-\sqrt{n}=\frac{d}{\sqrt{n+d}+\sqrt n}$
Infinite nested radical
$y=\sqrt{a+\sqrt{a+\cdots}}\ \Rightarrow\ y^{2}=a+y$
Rationalising a binomial surd denominator
$\frac{1}{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}$
Telescoping surd series
$\sum_{k=1}^{n}\frac{1}{\sqrt k+\sqrt{k+1}}=\sqrt{n+1}-1$
Exponential equation as a quadratic
$a^{2x}-S\,a^{x}+P=0\ \Rightarrow\ y=a^{x},\ y^{2}-Sy+P=0$