How to do Radicals

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

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Multiplying rule for Radical :

The following rule can be used to multiplying the radicals,

√a. √b= √(ab)

The most important thing when we multiply radicals is, both radical must have the common index.

For example,

√7 .√5 =√35
We can also write it as,

(7)(1/2). (5)(1/2)= (35)(1/2)
In the above example both having index 2.So that we can able to multiply directly.

Consider the another example,

(7)(1/3) . (5)(1/2)

We can not multiply the above radicals. Because both having different index.

Example Problems:

Learn radical problem 1:

Multiply the radicals:  √16 and √8

√16 √8 = √ (16 × 8)

= √ (4 × 4 × 4 × 2)

= 4 √ (4 × 2)

= 4 × 2 √2

= 8√2

Answer for the given learn radical problem is   8√2

Learn radical problem 2:

Multiply the radicals:  2√10 and  3√15

2√10.  3√15 = (2 × 3) √ (10 × 15)

= 6 √ (2 × 5 × 3 × 5)

= (6 × 5) √ (2 × 3)

= 30 √6

= 30 √6

Answer for the given learn radical problem is   30 √6

Learn radical problem 3:

Multiply the radicals: (√2x) (3√5x)

(√2x) (3√5x) = 3 √ (2x × 5x)

= 3√ (10x2)

= 3 x √10

Answer for the given learn radical problem is 3 x √10

Learn radical problem 4:

Multiply the radicals :  (3√9) and (`root(3)(125)`)

(3√9) . (`root(3)(125)`) =3 √( 3 × 3)(`root(3)(125)`)            

We know,   (`root(3)(125)`) = 5

=3 × 3 × 5

=45

Answer for the given learn radical problem is 45

Learn radical problem 5:

Divide the radicals:  √50 / √4

√50 / √4     = [√ (2 × 5 × 5)] / [√ (2 × 2)]

= [5√2] / [√ (2 × 2)]

= 5 /√2

Answer for the given learn radical problem is 5 /√2

Learn radical problem 6:

Simplify the radicals:  √18 and √2

√18 √2 = (√ (2 × 3 × 3 × 2)

= 3 × 2

= 6

Answer for the given learn radical problem is 6

Learn radical problem 7:

Simplify the radicals:   (√56) (9√25)

(√56) (9√25) = (√2×4×7).  9√25)

= 2(√14).(9 × 5)

= 90√14

Answer for the given learn radical problem is 90√14

Learn radical problem 8:

Simplify the radicals: √256 / √64

√256 / √64 = 16 / 8

= 2

Answer for the given learn radical problem is 2

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Learn radical problem 9:

Simplify the radicals: √ 12x3.  3√5

√ 12x3.  3√15 = (√ 3 × 4x3).(3√15)

=(2x√ 3x).(3√15)

=6x √ (9 × 5x)

= 6x × 3 √ (5x)

= 18x√ (5x)

Answer for the given learn radical problem is 18x√ (5x)

Learn radical problem 10:

Simplify the radicals:  9√25 / 3

9√25 / 3 = (9 × 5) / 3

=45 /3

= 15

Answer for the given learn radical problem is 15