Radicals in algebra are defined as the roots of a number or a variable. It is denoted with a symbol like v. The number which is mentioned in the ‘v’ portion of the symbol is called as the index of the symbol and the number or the variable or an expression entered inside the symbol is called radicand.
The meaning is that if the radicand is multiplied by itself index number of times, the product is the number that represents the radicand. If the index is 2, then the radical is called a square root and if the same is 3, then it is referred as cube root.
Usually for square roots, the index 2 is not mentioned. Thus radicals a type of expressions and like any other expression they can also undergo basic operations.
A radical can also be expressed in equivalent exponential form, with a rational exponent.. In general, nth root of am is equivalent to am/n. This concept is used in dividing radicals with variables. For example, (x^2)/v(x) can be simplified as (x^2)/(x1/2), which is, as per exponential rules is equal to (x3/2). This can be switched back to the radical form and can be expressed as v(x^3).
Let us discuss how do you divide radicals in general. Consider a radical in rational form. The division is possible if the indices of both the numerator and the denominator are same. In such a case, we can express the division of the radicands inside one symbol of the same index.
For example, v(a)/v(b) = v(a/b), b ? 0. In many cases it is possible to simplify the fraction (a/b) and at times it may turn out to a perfect power and hence the answer may be an integer free from radical sign.
Let us explain the concept of dividing radical with some examples.
v(3)/v(7) = v(3/7). In this case, the fraction cannot be simplified and this is the final answer.
v(14)/v(7) = v(14/7) . In this case, the fraction can be simplified as 2 and the final answer is v(2).
Now let us consider this one. v(28)/v(7) = v(28/7). The fraction 28/7 can be simplified as 4, which is a perfect square of 2. Therefore, in this case the answer turns out to be an integer.
There is one more concept in radical divisions. Suppose you arrive at an answer with a radical in the denominator. Never leave it in that form.
Multiply both the numerator and the denominator by the a radical which will make the denominator in integer form. This is called rationalizing the denominator.
The meaning is that if the radicand is multiplied by itself index number of times, the product is the number that represents the radicand. If the index is 2, then the radical is called a square root and if the same is 3, then it is referred as cube root.
Usually for square roots, the index 2 is not mentioned. Thus radicals a type of expressions and like any other expression they can also undergo basic operations.
A radical can also be expressed in equivalent exponential form, with a rational exponent.. In general, nth root of am is equivalent to am/n. This concept is used in dividing radicals with variables. For example, (x^2)/v(x) can be simplified as (x^2)/(x1/2), which is, as per exponential rules is equal to (x3/2). This can be switched back to the radical form and can be expressed as v(x^3).
Let us discuss how do you divide radicals in general. Consider a radical in rational form. The division is possible if the indices of both the numerator and the denominator are same. In such a case, we can express the division of the radicands inside one symbol of the same index.
For example, v(a)/v(b) = v(a/b), b ? 0. In many cases it is possible to simplify the fraction (a/b) and at times it may turn out to a perfect power and hence the answer may be an integer free from radical sign.
Let us explain the concept of dividing radical with some examples.
v(3)/v(7) = v(3/7). In this case, the fraction cannot be simplified and this is the final answer.
v(14)/v(7) = v(14/7) . In this case, the fraction can be simplified as 2 and the final answer is v(2).
Now let us consider this one. v(28)/v(7) = v(28/7). The fraction 28/7 can be simplified as 4, which is a perfect square of 2. Therefore, in this case the answer turns out to be an integer.
There is one more concept in radical divisions. Suppose you arrive at an answer with a radical in the denominator. Never leave it in that form.
Multiply both the numerator and the denominator by the a radical which will make the denominator in integer form. This is called rationalizing the denominator.
