# power, indices and surds Topic-wise Practice Test, Examples With Solutions & More Shortcuts

#### power, indices and surds & IT'S TYPES

Useful for Management (CAT, XAT, MAT, CMAT, IIFT, SNAP & other), Bank (PO & Clerk) SSC (CGL, 10+2, Steno, FCI, CPO, Multitasking), LIC (AAO & ADO) CLAT, RRB, UPSC and Other State PSC Exams

#### model 1 find largest and smallest value

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#### model 2 based on simplification

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#### model 3 based on positive and negative exponent

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#### model 4 simplifying roots with values

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#### model 5 simplifying roots of roots

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## Power, Indices And Surds - Basic Formulas, Shortcuts, Rules, Tricks & Tips - Quantitative Aptitude

#### Useful For All Competitive Exams Like UPSC, SSC, BANK & RAILWAY

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### Introduction to Power, Indices And Surds:

If a is a positive rational number, and n is a positive integer such that \$^n√a\$ or \$a^{1/n}\$ is irrational, then n√a is called a surd of order n. \$^n√a\$ is called the radical sign.

In \$^n√a\$, a is the radicand, and n is the order of the surd, also known as radical index. Surds are those quantities whose roots cannot be exactly obtained.

A pure surd is a surd with unity as its rational factor, the other factor being irrational, e.g., √7, \$^3\$\$√5\$, \$^4√{18}\$.

A mixed surd is a surd in which there is a rational factor other than unity and the irrational factor, e.g., 2√7, \$3√5\$, \$3^3√{18}\$.

### What is Rationalising Surds?

Rationalisation is the process of removing the radicals, i.e., roots (√) from an expression or a part of it without changing the value of the whole expression.

In a given fraction in the form of roots, we have to try and change the root form of the denominator by multiplying the denominator with a suitable quantity such that it becomes an integer.

If \$^n√a × ^n√b\$ is a rational number, then each of the two surds is known as a rationalising factor of the other.

If we multiply 3√8 by √8 , we get 3√8 × √8 = 3 × 8 = 24, a rational number. So √8 is the rationalising factor of 3√8 .

If the surd is of the type, a + √b , its rationalising factor is a – √b.

Rationalising factor of,

(i) \$1/{√a}\$ is √a

(ii) \$1/{a ± √b}\$ is a ± √b

(iii) \$1/{√a ± √b}\$ is √a ± √b

### What is Comprising Surds?

Surds can be compared only if they are of the same order. The radicals can then be compared.

• Comparing \$^3\$\$√4\$ and \$^3\$\$√8\$ , we may say that since 8 > 4,

\$^3\$\$√8 > ^3√4\$

• If the surds are of different order and different base, we need to reduce them to the same order.

Take \$^3\$\$√6\$ and \$^5\$\$√4\$

These terms are \$6^{1/3}\$ and \$4^{1/5}\$

We need to find the LCM of 3 and 5. LCM of 3 and 5 is 15.

Now, \$6^{1/3} = 6^{5/15} = {(6^{5})^{1/15}} = (7776)^{1/15}\$

and \$4^{1/5} = 4^{3/15} = {(4^{3})^{1/15}} = (64)^{1/15}\$

We may conclude \$^3√6 > ^5√4\$ [We need not even calculate 65 , as obviously \$6^5 > 4^3\$ .]

• You may be asked to arrange a set of surds in ascending/descending order. Take \$^4√6\$, √2 and \$^3√4\$.

### Important Mathematical Signs and Symbols with Explanation

Signs and Symbols Explanation
+ Plus, the sign of addition, e.g., 5 + 3. It also denotes a positive quantity, e.g., +3.
- Minus, the sign of subtraction. It also denotes a negative quantity
x Sign of multiplication.
÷ Sign of division.
. Dot at the centre of the two numbers is the sign of multiplication.
. Dot at the base of the two numbers is the sign of decimal.
= The sign of equality, read as equal to.
The sign of not equal to.
> Sign of greater than.
Sign of greater than or equal to.
< Sign of less than.
Sign of less than or equal to.
Sign of square root or under root.
\$^3\$\$√\$ Sign of cube root.
\$^n\$\$√\$ Sign of ‘n’th root.
\$a^2\$ The square or the second power of a; \$a^2 = a × a.\$
\$a^3\$ The cube or the third power of a; \$a^3\$ = a × a × a.
\$a^n\$ The 'n'th power of a.
| | Two vertical bars denote the absolute value of a number or mode of a number, e.g., | – 4 | = 4.
Sign of infinity
( ) Sign of parenthesis.
[ ] Sign of bracket.
{ } Figured bracket.
Sign of similarity.
Sign of congruency.
Sign of since or because.
Sign of therefore or hence.
Sign of union.
Sign of intersection.
Sign of subset.
Sign of 'belongs to' or 'is a member of'.
Sign of summation.
Sign of integration.
\$d/{dn}\$ Sign of differentiation.
Sign of identical to.
! Sign of factorial.
i Sign of \$√{−1}\$ .
? Question mark.

### "5" - Important Aptitude Rules, Formulas & Quick Tricks to Solve Power, Indices And Surds Based Problems

In this list of rules, you will get an idea that How to solve all different types & kinds of Power, Indices And Surds based aptitude problems asked in various competitive exams like UPSC, SSC, Bank, and Railway examinations at all levels.

By using this method, you can able to solve all problems from basic level to advanced level of questions asked based on Power, Indices And Surds in a faster approch.

Let's discuss the rules one by one with all Power, Indices And Surds Rules & Formulas with examples,

RULE 1 :

If any number is multiplied by the same number 'n' times, then,

a × a × a × a ... × a (n times ) = \$a^n\$

(i) where n and a are real numbers. (including fractions)
(ii) a is called base.
(iii) n is called indices.

RULE 2 :

\$a^m ×a^n= a^{m+n}\$

and \$a^m ×a^n×a^p= a^{m+n+p}\$

While multiplying. If base is same then powers get added.

RULE 3 :

While multiplying, if bases are different but powers are same then,

\$a^x ×b^x×c^x=(abc)^x\$

RULE 4 :

While dividing, if base is same then powers get subtracted, as

\$a^m÷a^n=a^{m-n}\$

RULE 5 :

If there is negative indices on a number, then

\$a^{-m}=1/a^m\$ or, \$a^m=1/a^{-m}\$

RULE 6 :

If there are indices on indices, then indices are multiplied. as–

(i) \$(a^m)^n=a^{mn}\$

(ii) \$(a^m)^{1/n}=a^{m/n}\$

(iii) \$((a^m)^n)^p=a^{mnp}\$

RULE 7 :

(i) \$a^{m^{n}}≠(a^m)^n\$

(ii) \$a^{m^{1/n}}≠(a^m)^{1/n}\$

(iii) \$a^{m^{{{n}}^p}}≠((a^m)^n)^p\$

RULE 8 :

Indices as fraction.

(i) \$(a/b)^m=a^m/b^m\$

(ii) \$(a/b)^{-m}=(b/a)^m\$

RULE 9 :

If \$a^x= a^y\$ then x = y and if \$x^n= y^n\$ then x = y

RULE 10 :

If the indices on any number is zero, the value of that number is 1, as

x° = 1, 5° = 1, 10° = 1, (50000)° = 1

RULE 11 :

If 'a' is a rational number and n is a positive integer, then, nth root of 'a', \$a^{1/n}\$ or \$√^n{a}\$ is an irrational number, \$√^n{a}\$ is called the surd of n indices, it means \$√^n{a}\$ is a surd where,

(i) 'a' is a rational number.

(ii) 'n' is a positive integer.

(iii) \$√^n{a}\$ is an irrational number.

RULE 12 :

If \$√^n{a}\$ is a surd, then n is called surd indices and a is called 'Radicand'. Every surd can be an irrational number, but every irrational number can not be a surd.

RULE 13 :

Mixed Surds : A surd having a rational co–efficient other than unity is called a mixed surd.

RULE 14 :

Pure Surd : The surds whose one factor is 1 and other factor is an irrational number, then that type of surd is called pure surd or the surd which is completely under radical sign.

RULE 15 :

Similar Surds : The surds whose irrational factor is same, that is called similar surds.

RULE 16 :

Irrational numbers as – \$√2, √3, √5, √7\$...etc. have infinite recurring decimals.

RULE 17 :

\$√^n{a}=(a)^{1/n}\$

RULE 18 :

\$(√^n{a})^n=a\$

RULE 19 :

\$√^n{ab}=√^n{a}×√^n{b}=(a)^{1/n}×(b)^{1/n}\$

RULE 20 :

\$√^n{√^n{a}}=((a)^{1/n})^{1/n}=a^{n^{1/2}}\$

RULE 21 :

\$√^n{a/b}=√^n{a}/√^n{b}=(a/b)^{1/n}\$

RULE 22 :

\$√^m{√^n{a}}=√^{mn}{a}\$

RULE 23 :

\$√{x√{x√{x...n times}}}= x^(1-1/{x^n})\$

RULE 24 :

If \$√{x-√{x-√{x-...∞}}}\$ where, x=n(n + 1) then

\$√{x-√{x-√{x -...∞}}}\$ = n

RULE 25 :

If \$√{x+√{x+√{x+...∞}}}\$ where, x=n(n + 1) then

\$√{x+√{x+√{x +...∞}}}\$ = (n + 1)

RULE 26 :

To find smallest or greatest out of these, we should equate all the indices and compare the base.

\$√^a{b},√^x{y},√^n{m},√^p{q}\$

### 5 - Types of Power, Indices And Surds Based Aptitude Questions and Answers Practise Test

Click the below links & Learn the specific model from Power, Indices And Surds problems that you have to practice for upcoming examination

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