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April 27, 2021

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Indices, Surds & Logarithms

This is aptitude questions and answers section on Indices and Surds, Logarithms with explanation for various interview, competitive examinations and entrance tests. This Indices and Surds, Logarithms section will linearly deal with some fundamental questions of Indices and Surds, Logarithms Quiz which are majorly asked in competitive exams like CAT, XAT, IBPS and government exams. Candidates who are attempting Indices and Surds, Logarithms Online test they can know their performance levels.

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Indices and Surds, Logarithms Questions – Indices and Surds, Logarithms Quiz Details

Online Test Name	Indices and Surds, Logarithms
Exam Type	Chapters & Multiple Choice Questions
Category	Aptitude Quiz
Number of Questions	30

Important Formulas – Surds and Indices

Laws of Indices

$a^{n} = a . a . a \dots (n times)$

$a^{m} . a^{n} . . . a^{p} = a^{m + n + . . . + p}$

$a^{m} . a^{n} = a^{m + n}$

$\frac{a^{m}}{a^{n}} = a^{m - n}$

${(a^{m})}^{n} = a^{m n} = {(a^{n})}^{m}$

${(a b)}^{n} = a^{n} b^{n}$

${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$

$a^{- n} = \frac{1}{a^{n}}$

$a^{n} = \frac{1}{a^{- n}}$

$a^{0} = 1$ where $a \in R, a \neq 0$

$a^{p / q} = \sqrt[q]{a^{p}}$

if $a^{m} = a^{n}$ where $a \neq 0$ and $a \neq \pm 1,$ then $m = n$

if $a^{n} = b^{n}$ where $n \neq 0,$ then $a = \pm b$

Surds

Let $a$ be a rational number and $n$ be a positive integer such that $\sqrt[n]{a}$ is irrational. Then $\sqrt[n]{a}$ is called a surd of order $n .$

Examples:

$\sqrt{3}$ is a surd of order $2$
$\sqrt[3]{7}$ is a surd of order $3$
$4 \sqrt{3}$ is a surd of order $2$

Please note that numbers like $\sqrt{9}$ , $\sqrt[3]{27}$ etc are not surds because they are not irrational numbers.

Every surd is an irrational number. But every irrational number is not a surd. (eg : $π$ , $e$ etc are not surds though they are irrational numbers.)

Quadratic Surds

A surd of order $2$ is called a quadratic surd

Examples

$\sqrt{2}, \sqrt{3}, (3 + \sqrt{5}),$ etc.

Rules of Surds

\sqrt{m} \times \sqrt{n} = \sqrt{m n}

$\frac{\sqrt{m}}{\sqrt{n}} = \sqrt{\frac{m}{n}}$

$\sqrt{a . \sqrt{a . \sqrt{a \dots n times}}} = a^{1 - \frac{1}{2^{n}}}$

$\sqrt{a . \sqrt{a . \sqrt{a \dots \infty}}} = a$

if $\sqrt{a + \sqrt{a + \sqrt{a \dots \infty}}} = p,$ then $p (p - 1) = a$

The conjugate surd of $\sqrt{a} + \sqrt{b} = \pm (\sqrt{a} - \sqrt{b})$

if $\sqrt{b}$ and $\sqrt{d}$ are quadratic surds and if $a + \sqrt{b} = c + \sqrt{d}$ ,
then $a = c$ and $b = d$
That is, rational part on left side = rational part on right side.
and irrational part on left side = irrational part on right side.

if $\sqrt{b}$ and $\sqrt{d}$ are quadratic surds and if $a + \sqrt{b} = \sqrt{d}$ ,
then $a = 0$ and $b = d$

Indices and Surds, Logarithms Aptitude Questions

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Indices and Surds, Logarithms MCQ Quiz Answers with Solutions

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Indices, Surds & Logarithms

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Indices and Surds, Logarithms Questions – Indices and Surds, Logarithms Quiz Details

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Important Formulas – Surds and Indices

Laws of Indices

Surds

Examples:

Quadratic Surds

Examples

Rules of Surds

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Indices and Surds, Logarithms Aptitude Questions

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Indices and Surds, Logarithms MCQ Quiz Answers with Solutions

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