The square root of a number is the value such that, when a number multiplied by itself, for example
3 x 3 = 9
It is written with a square root symbol " √ " and the number or expression inside the square root symbol is called the radicand.
Operations with square roots :
Addition, subtraction, multiplication and division of root terms can be performed by some laws. Let us see the rules one by one.
Rule 1 :
Whenever we have two or more root terms which are multiplied with same index, then we can put only one radical and multiply the terms inside the root terms.
Rule 2 :
Whenever we have two or more root terms which are dividing with same index, then we can put only one root and divide the terms inside the root sign.
Rule 3 :
nth root of a can be written as a to the power 1/n. Whenever we have power to the power, we can multiply both powers.
Addition and subtraction of two or more root terms can be performed with like radicands only. Like radicand means a number inside root sign must be same but the number outside the root sign may be different.
For example, 5√2 and 3√2 are like terms. Here the numbers inside the roots are same.
If we want to add, subtract, multiply or divide two or more root terms, the order must be same.
If the order of the radical terms are not equal, then we have to convert them with same order and we can perform multiplication or division.
Let us see some examples based on the above concepts.
Example 1 :
Simplify the following
4√3, 18√2, -3√3, 15√2
= 4√3 + 18√2 - 3√3 + 15√2
To simplify the above terms, we need to combine the like terms
= 4√3 - 3√3 + 18√2 + 15√2
= (4 - 3) √3 + (18 + 15) √2
= 1√3 + 33√2
= √3 + 33√2
Example 2 :
Simplify the following
2∛2, 24∛2, - 4∛2
= 2∛2 + 24∛2 - 4∛2
= (2 + 24 - 4) ∛2
= 22 ∛2
Example 3 :
Multiply ∛13 x ∛5
= ∛13 x ∛5
Since the index of both root terms are same, we can write only one root sign and multiply the numbers.
= ∛(13 x 5)
Example 4 :
Multiply 15√54 ÷ 3√6
= 15√54 ÷ 3√6
Since the index of both root terms are same, we can write only one root and divide the numbers.
= 5√9 ==> 5√(3 x 3) ==> 5 x 3 ==> 15
Example 5 :
Multiply (48)1/4 ÷ (72)1/8
= (48)1/4 ÷ (72)1/8
Since the index of the above root terms are not same, we need to convert the power 1/4 as 1/8.
= (48)(1/4) x (2/2) ÷ (72)1/8
= (48)(2/8) ÷ (72)1/8
= 482 (1/8) ÷ (72)1/8
= [(48 x 48) ÷ (72)]1/8
= [2304 ÷ 72]1/8
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