Multiplication is one of the basic operations in math. In arithmetic we multiply numbers whereas in algebra we multiply variables and expressions. Radicals are one type of expressions and they are also called roots with indices.The index of a square root is 2 but it is generally not indicated in the symbol.Let us discuss the concept of multiplying radicals. I like to share this Rules of Radicals with you all through my article.
The multiplication of expressions which are not in radical form is always defined. But it is not the case in radicals multiplication. There are certain rules for multiplying radicals for the multiplication to be defined. The most fundamental and the most important rule is that multiplication radicals are defined only if the indices of the radicals are same. A square root can be multiplied only with another square root and not with a cube root. Secondly, the radicands of radicals with even number indices cannot be negative.
With the above restrictions we can proceed to see how multiplication of roots is done.When the radical indices are same, the radical symbol can be ‘factored out’. That is, the radicands can be multiplied under one radical symbol. This is a great advantage and it makesthe multiplication simpler. In many cases the result may turn out to be an integer. I have recently faced lot of problem while learning Product Rule for Radicals, But thank to online resources of math which helped me to learn myself easily on net.
For example, consider the multiplication of √(8) by√(2). Both of them are irrational numbers. But as per the concept we explained, √(8)*√(2) = √(8*2) = √(16) = 4, which is an integer. Even in cases where the final answer may not be integers, we are still supposed to simplify the final radicand by factoring.
For example, √(6)*√(2) = √(6*2) = √(12). Though √(12) is irrational, 12 can be factored as 4*3 and 4 being a perfect square, it can be taken out. Thus the correct way to work it out is, √(6)*√(2) = √(6*2) = √(12) = √(4*3) = √(4)*√(3) = 2√(3).
The same concept is used in case of multiplying radicals with exponents, especially when variables are involved. Let us illustrate as to how it works.
√(x3)*√(x)= √(x3*x) = √(x4) = x2.
Even in cases where the radical symbol cannot be avoided, we still should try to keep the minimum exponent inside the symbol. For example,
√(x5)*√(x3) = √(x5*x3) = √(x15) = √(x14*x) = √(x14)*√(x) = x7*√(x)
As mentioned earlier, radicals of even number indices having negative radicands are not real numbers. Hence the multiplications in such cases have to be done by special techniques using the concept of imaginary numbers.
All imaginary numbers can be factored with √(-1) to remove the imaginary part and the letter ‘i’ is used as a symbol for √(-1).
The multiplication of expressions which are not in radical form is always defined. But it is not the case in radicals multiplication. There are certain rules for multiplying radicals for the multiplication to be defined. The most fundamental and the most important rule is that multiplication radicals are defined only if the indices of the radicals are same. A square root can be multiplied only with another square root and not with a cube root. Secondly, the radicands of radicals with even number indices cannot be negative.
With the above restrictions we can proceed to see how multiplication of roots is done.When the radical indices are same, the radical symbol can be ‘factored out’. That is, the radicands can be multiplied under one radical symbol. This is a great advantage and it makesthe multiplication simpler. In many cases the result may turn out to be an integer. I have recently faced lot of problem while learning Product Rule for Radicals, But thank to online resources of math which helped me to learn myself easily on net.
For example, consider the multiplication of √(8) by√(2). Both of them are irrational numbers. But as per the concept we explained, √(8)*√(2) = √(8*2) = √(16) = 4, which is an integer. Even in cases where the final answer may not be integers, we are still supposed to simplify the final radicand by factoring.
For example, √(6)*√(2) = √(6*2) = √(12). Though √(12) is irrational, 12 can be factored as 4*3 and 4 being a perfect square, it can be taken out. Thus the correct way to work it out is, √(6)*√(2) = √(6*2) = √(12) = √(4*3) = √(4)*√(3) = 2√(3).
The same concept is used in case of multiplying radicals with exponents, especially when variables are involved. Let us illustrate as to how it works.
√(x3)*√(x)= √(x3*x) = √(x4) = x2.
Even in cases where the radical symbol cannot be avoided, we still should try to keep the minimum exponent inside the symbol. For example,
√(x5)*√(x3) = √(x5*x3) = √(x15) = √(x14*x) = √(x14)*√(x) = x7*√(x)
As mentioned earlier, radicals of even number indices having negative radicands are not real numbers. Hence the multiplications in such cases have to be done by special techniques using the concept of imaginary numbers.
All imaginary numbers can be factored with √(-1) to remove the imaginary part and the letter ‘i’ is used as a symbol for √(-1).
