Exponents and Roots

Review ;  Exponents , Roots , Rules of Exponents

Exponents  

Exponents  refers  to  the repeated multiplication of a number by itself.

Like ;

2²  =  2 x 2  = 4

5⁶  = 5 x 5 x 5 x 5 x 5 x 5  = 15625

1⁸ =  1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 = 1

2,5 and 1  are base while 2,6 and 8 are exponents. We read it like 2 is raised to power of 2, 5 is raised to power of 6 and 1 is raised to power of 8

When exponent is 2 we refer it as square of base number.

3²  =  3 x 3  = 9

4²  = 4 x 4   = 16

5²  = 5 x 5  =  25

6²  = 6 x 6  = 36

(-2)²  =  (-2) x (-2) = 4

(-3)² = ( – 3) x ( -3) = 9

Similarly when exponent is 3 we refer it as cube of base number.

2³  = 2 x 2 x 2  =  8

3³  = 3 x 3 x 3  = 27

4³   =  4 x 4  x 4  = 64

(-2)³ = ( -2) x (-2) x (-2 )  = -8

 Basic Rules of Exponents

1.    X ⁰ = 1     where,  x ≠ 1
2.    0 ⁰ = undefined
3.    (-X)^a  = when ‘a’ is even   the value will be positive like

( -5 ) ² = (-5) x (-5) = 25

4.   (-X)^a  = when ‘a’ is odd the value will be negative like

(-2)³ = ( -2) x (-2) x (-2 )  = -8

 5.    X^-a =  1/ X^a   like

                      6^-2 =  1/6²  = 1 / 36

Roots

Square Root is a number of the value X² = a , being represented with a symbol ‘√’.

Like 3 is square root of 9 i.e. √9 = √(3 x3) = √(3)² = 3.

Square Root is an exponent with value of ½. A base is raised the power of ½. Each positive number has two square roots, one is positive and one is negative.

Like Square root of 4 is +2 and -2.

In real number system square root of negative number is undefined; like √-x = Undefined

Basic rules of Square Roots

• (√m)² = m                       m  > 0
• √m² = m              m  > 0

• For odd order roots like ‘3’ we can have only one solution for every number ‘m’, even if ‘m’ is negative.
• For even order roots like  2, 4, we will  have two solutions ; one is positive and one is negative for every positive number ‘m’, but NO ROOTS for any negative number ‘m’.

Examples ;

      9.    √ 25   =   5

      10.   √ – 25   =  Undefined