# Square Root and Cube Root 1 to 30

Importance : Questions based on square roots and cube roots are mainly asked with the questions of simplification and number system.

Scope of questions : Questions may be basic(totally numeric) or applied.

Way to success : Learning Formulae and squares/square roots/cube roots of different number is very useful.

Some important Points (On Square Roots) :

If a number n is multiplied with itself, then product n2 is called the Square of n and here n is called the Square root of n2.

If a number has x digits, then its square has (2x-1) digits.

Number is 12, Square is 144

∴ Number of digit in 144 is 2 X 2 – 1 = 3

If we square any number, then 2,3,7 and 8 will never come at unit place of square.

The square root of a negative number is always imaginary.

Square of two-digit number whose unit place digit is 5 can be obtained as.

(25)2 = 2 X 3 (Hundred) +52 = 2 X 300 + 25  = 625

or,

(35)2 = 3 X 4 (Hundred) + 52 = 3 X 400 + 25  = 1225

or,

(45)2 = 4 X 5 (Hundred) + 52 = 4 X 500 + 25 = 2025

There are two methods of calculating square root.

(i) Factor method

(ii) Division method

(i) Factor method : Square root of 44100

∴ 44100 = 2 X 2 X 3 X 3 X 5 X 5 X 7 X 7

$$∴\sqrt { 44100 } =\quad \sqrt { { 2 }^{ 2 }\times { 3 }^{ 2 }\times { 5 }^{ 2 }\times { 7 }^{ 2 } }$$

=2 X 3 X 5 X 7 = 210

(ii) Division method :  Square root of 455625

$$=\sqrt { 455625 } =\quad 675$$

Special Rules :

$$(i)\quad { \left( ab \right) }^{ \frac { 1 }{ 2 } }=\sqrt { ab } =\sqrt { a } \times \sqrt { b } ={ \left( a \right) }^{ \frac { 1 }{ 2 } }\times { \left( b \right) }^{ \frac { 1 }{ 2 } }$$

$$(ii)\quad { \left( \frac { a }{ b } \right) }^{ \frac { 1 }{ 2 } }=\frac { { \left( a \right) }^{ \frac { 1 }{ 2 } } }{ { \left( b \right) }^{ \frac { 1 }{ 2 } } } =\frac { \sqrt { a } }{ \sqrt { b } } =\sqrt { \frac { a }{ b } }$$

If the unit digit of a number is 1, then unit digit of its square root is 1 or 9 such as

$$\sqrt { 81 } =\quad 9\quad or,\quad \sqrt { 441 } =\quad 21$$

If the unit digit of any number is 4, then the unit digit of its square roots is 2 or 8. Such as

$$\sqrt { 64 } =\quad 8\quad or,\quad \sqrt { 144 } =\quad 12$$

If the unit digit of any number is 5 or 00(double zero), then the unit digit of its square is 5 or 0.

$$As,\sqrt { 625 } =\quad 25\quad or,\quad \sqrt { 100 } =\quad 10$$

Square roots of some numbers :

$$\sqrt { 0 } =0\quad \quad \quad \sqrt { 1 } =1$$

$$\sqrt { 2 } =1.414\quad \quad \quad \sqrt { 3 } =1.732$$

$$\sqrt { 4 } =2\quad \quad \quad \sqrt { 25 } =5$$

$$\sqrt { 9 } =3\quad \quad \quad \sqrt { 49 } =7$$

$$\sqrt { 16 } =4\quad \quad \quad \sqrt { 81 } =9$$

$$\sqrt { 36 } =6\quad \quad \quad \sqrt { 121 } =11$$

$$\sqrt { 64 } =8\quad \quad \quad \sqrt { 169 } =13$$

$$\sqrt { 100 } =10\quad \quad \quad \sqrt { 225 } =15$$

$$\sqrt { 144 } =12\quad \quad \quad \sqrt { 289 } =17$$

$$\sqrt { 196 } =14\quad \quad \quad \sqrt { 361 } =19$$

$$\sqrt { 256 } =16\quad \quad \quad \sqrt { 400 } =20$$

$$\sqrt { 324 } =18\quad \quad \quad \sqrt { 441} =21$$

$$\sqrt { 484 } =22\quad \quad \quad \sqrt { 529} =23$$

$$\sqrt { 576} =24\quad \quad \quad \sqrt { 625} =25$$

$$\sqrt { 676} =26\quad \quad \quad \sqrt { 729} =27$$

$$\sqrt { 784} =28\quad \quad \quad \sqrt { 841} =29$$

$$\sqrt {900} =30\quad \quad \quad \sqrt {961} =31$$

Some Important Points (On Cube Root) :

If a number n is multiplies by itself 3 times, then n3 is called the cube the cube of n and here n is called the cube root of n3 .

Cube roots can be calculated only by factor method.

If in any number 0,1,2,3,4,5,6,7,8 or 9 are at unit place, then 0,1,8,7,4,5,6,3,2 or 9 respectively will be unit place of their cube root.

Note that if unit place of nay number is 0,1,4,5,6 or 9 , then unit place of the cube or cube root of this number will be the same as in original number.

To calculate cubic root of 3375.

3375 = 3 X 3 X 3 X 5 X 5 X 5

$$∴\quad \sqrt [ 3 ]{ 3375 } =\sqrt [ 3 ]{ { 3 }^{ 3 }\times { 5 }^{ 3 } } =\quad 3\times 5\quad =\quad 15$$

$$If\quad \sqrt [ 3 ]{ x } =b\quad then\quad { x }^{ \frac { 1 }{ 3 } }=b$$

$$\Rightarrow \quad \log { { x }^{ \frac { 1 }{ 3 } } } =\quad logb$$

$$\Rightarrow \quad \frac { 1 }{ 3 } \log { { x } } =\quad logb$$

$$∴\quad b = antilog(\frac { 1 }{ 3 } \log { x } )$$

Some Cube Roots :

$$\sqrt [ 3 ]{ 1 } =1,\quad \sqrt [ 3 ]{ 8 } =2$$

$$\sqrt [ 3 ]{ 27 } =3,\quad \sqrt [ 3 ]{ 64 } =4$$

$$\sqrt [ 3 ]{ 125 } =5,\quad \sqrt [ 3 ]{ 216 } =6$$

$$\sqrt [ 3 ]{ 343 } =7,\quad \sqrt [ 3 ]{ 512 } =8$$

$$\sqrt [ 3 ]{ 729 } =9,\quad \sqrt [ 3 ]{ 1000 } =10$$

$$\sqrt [ 3 ]{ 1331 } =11,\quad \sqrt [ 3 ]{ 1728 } =12$$

$$\sqrt [ 3 ]{ 2197 } =13,\quad \sqrt [ 3 ]{ 2744 } =14$$

$$\sqrt [ 3 ]{ 3375 } =15,\quad \sqrt [ 3 ]{ 4096 } =16$$

$$\sqrt [ 3 ]{ 4913 } =17,\quad \sqrt [ 3 ]{ 5832 } =18$$

$$\sqrt [ 3 ]{ 6859 } =19,\quad \sqrt [ 3 ]{ 8000 } =20$$

$$\sqrt [ 3 ]{ 9261} =21,\quad \sqrt [ 3 ]{ 10648 } =22$$

$$\sqrt [ 3 ]{ 12167} =23,\quad \sqrt [ 3 ]{ 13824} =24$$

$$\sqrt [ 3 ]{ 15625} =25,\quad \sqrt [ 3 ]{ 17576} =26$$

$$\sqrt [ 3 ]{19683} =27,\quad \sqrt [ 3 ]{ 21952} =28$$

$$\sqrt [ 3 ]{ 24389} =29,\quad \sqrt [ 3 ]{27000} =30$$

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Simplification Formulas | SSC,IBPS,BANK,ARMY,GOVT EXAM

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1. Nice website