When a number is multiplied by itself, the product is called the square of that number.

The number itself is called the radical of the product.

That is,

√(3x3)  =  3

Based on the definition given above for radicals, let us look at the properties of radicals.

Property 1 :

Whenever we have two or more radical terms which are multiplied with same index, then we can put only one radical and multiply the terms inside the radical. Property 2 :

Whenever we have two or more radical terms which are dividing with same index, then we can put only one radical and divide the terms inside the radical. Property 3 :

If we have radical with the index "n", the reciprocal of "n", (That is, 1/n) can be written as exponent. And also,  whenever we have exponent to the exponent, we can multiply both the exponents. (This is one of the laws of exponents)

Property 4 :

Addition and subtraction of two or more radical terms can be performed with like radicands only. Like radicand means a number which is inside the radical must be same but the number outside the radical may be different.

For example, 5√2 and 3√2 are like radical terms. Here the numbers inside the radicals are same.

Property 5 :

If we take radical sign with the index "n" from one side of the equation to the other side of the equation, "n" will be at the exponent. Property 6 :

If the units digit of a number is 2, 3, 7 or 8, then the number can not be a perfect square. So the square root of those numbers will be irrational.

For example,

√23  =  4.795831.........................

Property 6 :

Property 7 :

If a number ends in an odd number of zeros, then, the square root of the number will be irrational.

√3000  =  54.772255.........................

Property 8 :

If a square number is followed by even number of zeros, square root of the number will be rational.

√40000  =  200

In the above result (That is 200), the number of zeros is half the number of zeros in the number inside the square root.

Property 9 :

The square root of an even perfect square number is always even and the square root of an odd perfect square number is always is odd.

For example,

√144  =  144

225  =  15

Property 10 :

Whenever we have negative number in the square root, then it is  called imaginary.

For example,

√(-9), √(-12) and √(-225)

## Properties of radicals - Simplification

To simplify a number which is in radical sign we need to follow the steps given below.

Step 1:

Split the numbers in the radical sign as much as possible

Step 2:

If two same numbers are multiplying in the radical, we need to take only one number out from the radical.

Step 3:

In case we have any number in front of radical sign already,we have to multiply the number taken out by the number which is in front of radical sign already.

Step 4:

If we have radical with the index n, (That is,  ) and the same term is multiplied by itself "n" times, then we need to take out only one term out from the radical.

For example, if we have radical with the index 3,(That is, ∛ ) and the same term is multiplied by itself three times, we need to take out only one term out from the radical.

## Properties of radicals - Practice problems

Problem 1:

Simplify the following √5  √18

Solution :

=  √5 ⋅ √18

According to the laws of radical,

=  √(5  18) ==> √(5  3  3  2)  ==> 3 √(5  2)  ==>  3 √10

Problem 2 :

Simplify the following ∛7  ∛8

Solution :

=  ∛7  ∛8

According to the laws of radical,

= ∛(7  8) ==> ∛(7  2  2  2) ==> 2 ∛(7  2) ==> 2 ∛14

Problem 3 :

Simplify the following 3√35 ÷ 2√7

Solution :

=   3√35 ÷ 2√7

According to the laws of radical,

=  (3/2) √(35/7) ==> (3/2)√5

Problem 4 :

7 √30 + 2 √75 + 5 √50

Solution :

= 7 √30 + 2 √75 + 5 √50

First we have to split the given numbers inside the radical as much as possible. =  7√(5   3) + 2√(5  5  3) + 5√(5  5  2)

Here we have to keep √30 as it is.

=  7√30 + (2 ⋅ 5) √3 + 5 √2

=  7√30 + 10 √3 + 5 √2

Problem 5 :

√27 + √105 + √108 + √45

Solution :

= √27 + √105 + √108 + √45

First we have to split the given numbers inside the radical as much as possible √27  =  √(3  3  3)  =  3√3

√105  =  √(5  3  7)  √105

√108  =  √(3  3  3  2  2)  =  (3  2)√3  =  6√3

√45  =  √(3  3  5)  =  3√5

√27 + √105 + √108 + √45  =  3√3 + √105 + 6√3 + 3√5

=  9√3 + √105 + 3√5

Problem 6 :

√45 + 3 √20 + √80 - 4 √40

Solution :

= √45 + 3 √20 + √80 - 4 √40

First we have to split the given numbers inside the radical as much as possible. √45 + 3 √20 + √80 - 4 √40

√45  =  √(3  3  5)  =  3√5

3 √20  =  √(2  2 ⋅ 5)  =  2√5

√80  =  √(2  2  2  2  5)  =  (2  2)√5  =  4√5

4√40  =  4√(2  2  2  5)  =  (4  2)√(5  2)  =  8√10

√45 + 3 √20 + √80 - 4 √40  =  3√5 + 2√5 + 4√5 - 8√10

=  (3 + 2 + 4)√5 - 8√10

=  9√5 - 8√10

Problem 7 :

3 √32 - 2√8 + √50

Solution :

=  3 √32 - 2 √8 + √50

First we have to split the given numbers inside the radical as much as possible. 3 √32  =  3 √(2  2  2  2  2)  =  (3  2  2)√2  =  12√2

2 √8  =  2√(2  2  2)  =  (2  2)√2  =  4√2

√50  =  √(5  5  2)  =  5√2

3 √32 - 2 √8 + √50  =  12√2 - 4√2 + 5√2

=  (12 - 4 + 5) √2

=  (17 - 4) √2  =  13√2

Problem 8 :

2 √12 - 3√27 - √243

Solution :

=  2 √12 - 3 √27 - √243

First we have to split the given numbers inside the radical as much as possible. 2 √12  =  2 √(2  2  3)  =  (⋅ 2) √3  =  4√3

3 √27  =  3 √(3  3  3)  =  (3 ⋅ 3) √3  =  9 √3

√243  =  √(3  3    3)  =  (3 ⋅ 3) √3  =  9 √3

2 √12 - 3 √27 - √243  =  4√3 - 9√3 - 9√3

=  (4 - 9 - 9)√3

=  -14√3

Problem 9 :

√54 - √2500 - √24

Solution :

=  √54 - √2500 - √24

First we have to split the given numbers inside the radical as much as possible. √54  =  √(3  3  3 ⋅ 2)  =  3√6

√2500  =  √(5  5  10 ⋅ 10)  =  5 ⋅ 10  =  50

√24  =  √(2  2  2  3)  =  2 √(2 3)  =  √6

√54 - √2500 - √24  =  3√6 - 50 - 2√6

=  √6 - 50

Hence the answer is √6 - 50.

Problem 10 :

Solve for "x" :

2√x - 2  =  10

Solution :

2√x - 2  =  10

2√x  =  12

Divide by "2" on both sides

√x  =  6

x  =  6²

x  =  36

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