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(male narrator) In this video,
we will look at how we can add radicals
if we at least have no like radicals to begin with.
The way we handle this
is before adding the radicals together,
we will reduce the radicals,
and hopefully, this will give us some like radicals to work with.
For example, in this problem, none of the radicals are alike.
However, we can simplify the radicals
by finding the prime factorizations
and dividing the exponents by the index.
We will assume you know how to find a prime factorization.
The prime factorization of 50 is 2 times 5 squared with an x;
plus 5 times the square root of 27, which is 3 cubed;
minus 3 times the square root of 2x, 2 is already prime;
minus 2 times the square root of 108.
Finding its prime factorization is 2 squared times 3 cubed.
We can now simplify it
by dividing the exponents by the index of 2;
5 squared can come out as a 5,
giving us 20 times the square root of 2x;
3 cubed has one 3 coming out; and we have a remainder of 1,
so one 3 will remain in the radical.
Multiplying, we now have 15 times the square root of 3;
minus 2 times the square root of 2x;
minus...we can pull one 2 out;
dividing the exponent by the index;
and we can also pull one 3 out.
There's a remainder of 1, so one 3 remains behind.
We now have 12... times the square root of 3.
We're now ready to combine like radicals.
Square root of 2x makes like radicals,
and 20 minus 2 means we have 18 of these square roots of 2x.
We also have like radicals on the root 3,
and 15 minus 12 is +3 of these square roots of 3.
This is our final solution.
Let's try another example
where we have to simplify the radicals first
before we can find any like radicals.
In this problem, we have cube roots,
but the process is identical.
We must find the prime factorizations of the numbers
to see if we can do any simplifying:
81 is 3 to the fourth power; x cubed y; minus 3y;
times the cube root of 32;
its prime factorization is 2 to the fifth;
x squared; plus x; times the cube root of 24y;
24's prime factorization is 2 cubed times 3y;
minus the cube root of 500.
Finding its prime factorization
by dividing out all the prime factors,
gives us 2 squared times 5 cubed;
with x squared y cubed.
We can now go back and simplify
by dividing all the exponents by the index
and leaving any remainders behind in the radical:
3 to the fourth;
dividing the exponent by 3 gives us 1;
and one 3 remains in the radical; x cubed;
dividing the exponent by 3 means one x can come out.
We now have 3x times the cube root of 3y;
minus 2 to the fifth.
When we divide that exponent by 3, we get 1,
but there's also a remainder of 2;
2 times 3 on the outside
gives us 6y times the cube root of 2 squared, or 4x squared;
plus 2 cubed;
dividing the exponent by 3 gives us 2 to the first;
giving us 2x times the cube root of 3y;
minus...whoops...5 cube pulls a 5 out; y cube pulls a y out.
We have 5y times the cube root of 2 squared, or 4x squared.
Notice, we have like terms: x cubed root of 3y;
5x cube root of 3y; adding the 3 and the 2 together;
and also -6 and -5 gives us -11
of these y cube root of 4x squares.
This is our final answer.