# Surd and indices in Mathematics

**Surds :**

Let x is a rational number(i.e. can be expressed in p/q form where q ≠ 0) and n is any positive integer such that x^{1/n} = ^{n }√x is irrational(i.e. can’t be expressed in p/q form where q ≠ 0), then that ^{n }√x is known as surd of nth order.

**Example –**

√2, √29, etc.

√2 = 1.414213562…, which is non-terminating and non-repeating, therefore √2 is an irrational number. And √2= 2^{1/2}, where n=2, therefore √2 is a surd. In simple words, surd is a number whose power is an infraction and can not be solved completely(i.e. we can not get a rational number).

**Indices :**

- It is also known as power or exponent.
- X
^{p}, where x is a base and p is power(or index)of x. where p, x can be any decimal number.

**Example – **

Let a number 2^{3}= 2×2×2= 8, then 2 is the base and 3 is indices.

- An exponent of a number represents how many times a number is multiplied by itself.
- They are used to representing roots, fractions.

**Rules of surds :**

When a surd is multiplied by a rational number then it is known as a mixed surd.

**Example –**

2√2, where 2 is a rational number and √2 is a surd. Here x, y used in the rules are decimal numbers as follows.

S.No. | Rules for surds | Example |
---|---|---|

1. | ^{n }√x = x^{1/n} |
√2 = 2^{1/2} |

2. | ^{n}√(x ×y) =^{n }√x × ^{n }√x |
√(2×3)= √2 × √3 |

3. | ^{n}√(x ÷y)=^{n }√x ÷ ^{n }√y |
^{3}√(5÷3) = ^{3}√5 ÷ ^{3}√3 |

4. | (^{n }√x)^{n} = x |
(√2)^{2} = 2 |

5. | (^{n}√ x)^{m} = ^{n}√(x ^{m}) ^{ } |
(^{3}√27)^{2} = ^{3}√(27^{2}) = 9 |

6. | ^{m}√(^{n}√ x) = ^{m × n }√x |
^{2}√(^{3}√729)= ^{2×3}√729 = ^{6}√729 = 3 |

**Rules of indices : **

S.No. | Rules for indices | Example |
---|---|---|

1. | x^{0 }= 1 |
2^{0} = 1 |

2 | x ^{m} × x ^{n} = x ^{m +n} |
2^{2} ×2^{3}= 2^{5} = 32 |

3 | x ^{m }÷ x ^{n} = x ^{m-n} |
2^{3} ÷ 2^{2} = 2^{3-2} = 2 |

4 | (x ^{m})^{n} = x ^{m ×n} ^{ } |
(2^{3})^{2} = 2^{3×2} = 64 |

5 | (x × y)^{n} = x ^{n} × y ^{n} |
(2 × 3)^{2}= 2^{2} × 3^{2} =36 |

6 | (x ÷ y)^{n} = x ^{n} ÷ y ^{n} |
(4 ÷ 2) ^{2}= 4^{2} ÷ 2^{2} = 4 |

**Other Rules :**

Some other rules are used in solving surds and indices problems as follows.

// From 1 to 6 rules covered in table.7) x^{m}= x^{n}then m=n and a≠ 0,1,-1.8) x^{m}= y^{m}thenx = y if m is evenx= y, if m is odd

**Basic problems based on surds and indices :**

**Question-1** :

Which of the following is a surd?

a)^{2}√36 b)^{5}√32 c)^{6}√729 d)^{3}√25

**Solution – **

An answer is** **an option (d)

Explanation -3√25= (25)^{1/3}= 2.92401773821... which is irrational So it is surd.

**Question-2 :**

Find √√√3

a) 3^{1/3}b) 3^{1/4}c) 3^{1/6}d) 3^{1/8}

**Solution –**

An answer is an option (d)

Explanation -((3^{1/2})^{1/2})^{1/2}) = 3^{1/2 × 1/2 ×1/2}= 3 1/8 according to rule number 5 in indices.

**Question-3 :**

If (4/5)^{3} (4/5)^{-6}= (4/5)^{2x-1}, the value of x is

a) -2 b)2 c) -1 d)1

**Solution – **

The answer is option (c)

Explanation -LHS=(4/5)^{3}(4/5)^{-6}=^{ }(4/5)^{3-6}= (4/5)^{-3}RHS = (4/5)^{2x-1}According to question LHS = RHS ⇒ (4/5)^{-3}= (4/5)^{2x-1}⇒ 2x-1 = -3 ⇒ 2x = -2 ⇒ x = -1

**Question-4 :**

3^{4x+1}= 1/27, then x is

**Solution –**

3^{4x+1}= (1/3)^{3}⇒3^{4x+1}= 3^{-3}⇒4x+1 = -3 ⇒4x= -4 ⇒x = -1

**Question-5 :**

Find the smallest among 2 ^{1/12,} 3 ^{1/72}, 4^{1/24},6^{1/36}.

**Solution – **

The answer is 3^{1/72}

**Explanation –**

As the exponents of all numbers are infractions, therefore multiply each exponent by LCM of all the exponents. The LCM of all numbers is 72.

2^{(1/12 × 72)}= 2^{6}= 64 3(^{1/72 ×72)}= 3 4^{(1/24 ×72)}= 4^{3}= 64 6^{(1/36 ×72)}= 6^{2}= 36

**Question-6 :**

The greatest among 2^{400, }3^{300},5^{200},6^{200}.

a) 2^{400}b)3^{300}c)5^{200}d)6^{200}

**Solution –**

An answer is an option (d)

**Explanation –**

As the power of each number is large, and it is very difficult to compare them, therefore we will divide each exponent by a common factor(i.e. take HCF of each exponent).

The HCF of all exponents is 100. 2^{400/100}= 2^{4}= 8. 3^{300/100}= 3^{3}= 27 5^{200/100}= 5^{2}= 25 6^{200/100}= 6^{2}= 36 So 6^{200}is largest among all.