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

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

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#### SSC EXAM BASED EXERCISE

## power, indices and surds Topic-wise Types, Definitions, Important fact & Techniques with Short Tricks & Tips useful for all competitive Examinations

## 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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