SOLUTION: how do you multilply radicals with different radicands and different radicals

Question 454521: how do you multilply radicals with different radicands and different radicals
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
how do you multilply radicals with different radicands and different radicals
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1. Within a radical, you can perform the same calculations as you do outside the radical.
2. With radicals of the same indices, you can also perform the same calculations as you do outside the radical, but still staying inside the radical(s).
3. So, what do you do with radicals of different indices.
Note that any radican can be written as an expression with a fractional exponent.
For example, √10 can be written as 10^1/2, cube root (7)=7^1/3, 4th root of 15=15^1/4,etc.
Notice that the denominator of the fractional exponent always equals the index.
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What if I took the √(10^3). In fractional exponent form this would be 10^2/3, cube root (7^5)=7^5/3, 4th root of 15^7=15^7/4, etc. Note that as we raise the power of the radican, the index does not change. In other words, the numerator of the fraction is the power or degree of the expression and the index is the denominator.
Example:(a) cube root(x^5)=x^5/3=(cube root(x))^5
(b)4th root(y^5)=y^5/4=(4th root(y))^5
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Now, let us get to your question:
say, we multiply;
cube root(x^5)*4th root(y^5)
we can't multiply them because they have different indices, but in fractional form they have exponents of 5/3 and 5/4. Why don't we change these fractions so they have a common denominator like 20/12 and 15/12. In the first case we multiplied the denominator by 4 so we also need to multiply the numerator by 4 to get 20/12. In the second case, you multiplied by 3 to get the LCD of 12. We now have two radicals with a common index of 12 but you also had to raise the power or degree of their radicans by their respective multipliers.
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This is what you should end up with:
12th root(x^20)*12th root(y^15)=12th root((x^20y^15)
A better example would be to use a polynomial instead of just x and y, but I hope this helps you understand how to work with radicals.
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