Like Radicals
Like radicals are the radicals that have the same index and same radicand. And those numbers which lies in the same root or those number whose power are same are called like Radicals.
For example:
$\sqrt{2}$, $\sqrt{3}$, $\sqrt{4}$, $\sqrt{5}$
Since all the radicals have same power $\frac{1}{2}$ for square root so these are called like radicals.
$\sqrt[3]{2}$, $\sqrt[2]{4}$, $\sqrt[6]{5}$ are not like radicals as their roots are not same as their indices are different although the radicands are same.
When two radicals have different radicand number, they are called unlike radicals. Some unlike radicals can be simplified and made them like radicals. Symbolicaly if n√x and n√y are two radicals then, if x = y, then they are like radicals otherwise it is called as unlike radicals.
We can add or subtract only Like Radicals.
Below are the examples on like radicals -
Solved Examples
Solution:
Take L.C.M. of = (3, 2, 5) = 30
Write (2)1/3 = 210/30
31/2 = 315/30
(5)1/5 = 56/30
(210)1/30,(315)1/30 , (56)1/307
$\sqrt[30]{2^{10}}$, $\sqrt[30]{3^{15}}$, $\sqrt[30]{5^6}$ all have same radicals.
Solution:
Takes L.C.D. of (2,3) = 6
Write (5)1/2 = 53/6 = (53)1/6
(7)1/3 = 52/6 = (72)1/6
= (125)1/6, (49)1/6
= $\sqrt[6]{125}$, $\sqrt[6]{49}$
Same radicals and now it can be easily compared Here we see that 125>49
There fore $\sqrt[6]{125}$ > $\sqrt[6]{49}$
(5)1/2 > (7)1/3
Below are some examples for solving like radicals
Adding Like Radicals
Problem 1: Adding like radicals $\sqrt{72}$ + $\sqrt{50}$
Solution: First we have to split the radical for each of the radical in above.
So that we have to find the perfect square for the each above and place the perfect square in first.
$\sqrt{36}$ x $\sqrt{}$ + $\sqrt{25}$ x $\sqrt{}$
Then, place other than perfect square in second.
$\sqrt{36}$ x $\sqrt{2}$ + $\sqrt{25}$ x $\sqrt{2}$
Square roots of the perfect square are taken.
6 x $\sqrt{2}$ + 5 x $\sqrt{2}$
Therefore these radicals are LIKE radicals, because we are combining like radicals.
11 x $\sqrt{2}$
Adding unlike radicals
Problem 2: Adding unlike radicals $\sqrt{125}$ + $\sqrt{50}$
Solution: First we have to split the radical for each of the radical in above.
So that we have to find the perfect square for the each above and place the perfect square in first.
$\sqrt{25}$ x $\sqrt{}$ + $\sqrt{25}$ x $\sqrt{}$
Then, place other than perfect square in second.
$\sqrt{25}$ x $\sqrt{5}$ + $\sqrt{25}$ x $\sqrt{2}$
Square roots of the perfect square are taken.
5 x $\sqrt{5}$ + 5 x $\sqrt{2}$
Therefore the given radicals are not LIKE radicals, because we can’t combine since there are no like terms.
Problem 3: Find whether the given terms are like radical terms or not?
i) - $\sqrt{5}$ and $\sqrt[3]{5}$
ii) -2$\sqrt{y}$ and $\sqrt{2x}$
iii) -x $\sqrt{5}$ and y $\sqrt{5}$
iv) -5 $\sqrt{x}$ and 5 $\sqrt{y}$
Solution:
i) - $\sqrt{5}$ and $\sqrt[3]{5}$
Here you have a square root and a *cube* root. The given expression is not a like radicals.
ii) -2$\sqrt{y}$ and $\sqrt{2x}$
The given terms are NOT like radicals, though you could factor out -2 from the second, then they would be.
iii) -x $\sqrt{5}$ and y $\sqrt{5}$
The given terms are LIKE radicals
iv) -5 $\sqrt{x}$ and 5 $\sqrt{y}$
The given terms are NOT a like radicals
