A radical of a number ‘x’ is defined as the principal root (say nth root) of ‘x’ and algebraically denoted as,
The letter ‘n’ is known as the index of the radical and ‘x’ is called as the radicand. Two radicals are said to be like radicals if their indices are same. If not those radicals are referred as unlike radicals. In n = 2, the radical is better known as square root and normally the index 2 is not denoted. That is, square root of 2 is simply denoted as √(2). If n = 3, then the radical is called as cube root but in this case the index 3 ought to be mentioned. I like to share this How to do Radicals with you all through my article.
√(2), √(5), √(6), √(7) are all like radicals examples.
If the indices and the radicands are same, then such principal roots can be considered as like terms and can act as common factors. This concept is used in combining like-radicals. Let us illustrate this with some examples.
5√(2) + 3√(2) = (5 + 3)√(2) = 8√(2)
5√(2) - 3√(2) = (5 - 3)√(2) = 2√(2)
In certain cases the like terms may not be readily seen. But by proper algebraic methods it may still be possible to combine. Look at the following example.
√(8) + √(18) = ?
We see that the indices are same but still we are unable to simplify. But the concept of factoring and the knowledge of square roots helps us to simplify.
√(8) + √(18) = √(4*2) + √(9*2) = √(4) * √(2) + √(9) * √(2) = 2 * √(2) + 3 * √(2) = (2 + 3)*√(2) = 5√(2)
Further, it may be noted that multiplication and division of like radicals can be performed easily. If the roots have same indices, then the product of two roots is same as the root of the product of the radicands. Same is the case for division of two roots. That is,
√(x)*√(y) = √(x*y) and √(x)/√(y) = √(x/y)
These concepts are very important and let us see how it works.
Example 1: √(27)*√(3) = ?
√(27)*√(3) =√(27*3) = √(81) = 9
Example 2: √(27)/√(3) = ?
√(27)/√(3) = √(27/3) = √(9) = 3
Sometimes, this concept is also helpful in simplification of unlike radicals. For example, cube root of 64 is same as the square root of 16, which is 4. Hence if there is problem like to combine cube root of 64 and square root of 121, one can easily figure out the answer as 4 + 11 = 15.
The letter ‘n’ is known as the index of the radical and ‘x’ is called as the radicand. Two radicals are said to be like radicals if their indices are same. If not those radicals are referred as unlike radicals. In n = 2, the radical is better known as square root and normally the index 2 is not denoted. That is, square root of 2 is simply denoted as √(2). If n = 3, then the radical is called as cube root but in this case the index 3 ought to be mentioned. I like to share this How to do Radicals with you all through my article.
√(2), √(5), √(6), √(7) are all like radicals examples.
If the indices and the radicands are same, then such principal roots can be considered as like terms and can act as common factors. This concept is used in combining like-radicals. Let us illustrate this with some examples.
5√(2) + 3√(2) = (5 + 3)√(2) = 8√(2)
5√(2) - 3√(2) = (5 - 3)√(2) = 2√(2)
In certain cases the like terms may not be readily seen. But by proper algebraic methods it may still be possible to combine. Look at the following example.
√(8) + √(18) = ?
We see that the indices are same but still we are unable to simplify. But the concept of factoring and the knowledge of square roots helps us to simplify.
√(8) + √(18) = √(4*2) + √(9*2) = √(4) * √(2) + √(9) * √(2) = 2 * √(2) + 3 * √(2) = (2 + 3)*√(2) = 5√(2)
Further, it may be noted that multiplication and division of like radicals can be performed easily. If the roots have same indices, then the product of two roots is same as the root of the product of the radicands. Same is the case for division of two roots. That is,
√(x)*√(y) = √(x*y) and √(x)/√(y) = √(x/y)
These concepts are very important and let us see how it works.
Example 1: √(27)*√(3) = ?
√(27)*√(3) =√(27*3) = √(81) = 9
Example 2: √(27)/√(3) = ?
√(27)/√(3) = √(27/3) = √(9) = 3
Sometimes, this concept is also helpful in simplification of unlike radicals. For example, cube root of 64 is same as the square root of 16, which is 4. Hence if there is problem like to combine cube root of 64 and square root of 121, one can easily figure out the answer as 4 + 11 = 15.
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