Prepare for Radicals

The radical symbol was introduced by John Bell in 1668.Radicals having the same properties as the properties of the numbers. A radical is an expression that expresses the square roots, cube roots etc. Radical is the word that comes from the Latin word Radix. When we use the radical without a sign or index number, it mentions the positive square root of the radicand.

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Explanation of prepare for radicals:

In radicals have square root, cubic root, and nth root. Square root has a root of degree 2.cubic root has a root of degree 3 and nth root has a root of degree n number. Roots are using the radical symbol,

root(n)(x) n = x

Square root:

To prepare the radical real number has the one positive root and one negative root, for example square roots of 36 are 6 and -6. The other name of positive square root is principal square root and mentioned the radical sign. √36 = 6

The negative numbers has square root of imaginary number is two. For example, square root of -36 is 6i and -6i

Here, i = √-1 or i2 = -1

Cubic roots:

The cubic root  x can be written as  root(3)(x)

For example,  root(3)(125) = 5  and   root(3)(-125) = -5

Identities and properties for prepare for radicals:

The basic relationship between the exponent and radical

root(n)(a)m  = ( root(n)(a) )m = (a1/n )m = am/n

√ -1 ×√-1 = -1 where as √(-1 × -1) = 1

Radical property for multiplication      root(n)(ab) = root(n)(a) root(n)(b) ,

Radical property for division          root(n)((a/b)) = root(n)(a)  /  root(n)(b)

Here,  a, b, ab, (a/b) are radicand

n - index

Examples:

Prepare for radical problem 1:

Multiply:  √-12 and √8

Solution:

(√-12) (√8) = √ - (12×8)

= √ (-96)

= √- (2 × 3 × 4 × 4).                We know √ (-1) = i,

= 4√ [(-1) 6]                                                 -1 = i2

so,            = 4i √6

Answer of given prepare for radical is 4i √6

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Prepare for radical problem 2:

Dividing the radicals: (√3x2 / 3√5x3)

Solution:

(√3x2 / 3√5x3) = (x√ 3) / x (3√ (5x))

= √ 3 / (3√ (5x))

=√ 3 / (√ 3√ 3 √ (5x))

= 1 / [√ 3√ (5 x)]

= 1 / [√ (15 x)]

Answer of given prepare for radical is 1 / [√ (15 x)]