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(male narrator) In this video,
we'll look at multiplying radicals of mixed index;
at first, by using rational exponents.
You may notice this first radical has an exponent of 3,
and so, we can write that as: a to the 2/3, b to the 1/3.
The second has a radical index of 4,
so we have: a to the 1/4, b to the 2/4.
In order to combine these together,
we would want to add the exponents together,
but before we can add exponents,
we need to have a common denominator.
Let's make everything have a common denominator of 12.
By multiplying by 4 over 4 on the first two fractions
and 3 over 3 on the second two fractions,
we now have: a to the 8/12, b to the 4/12,
a to the 3/12, and b to the 6/12.
So we would be able to add the exponents together
to get: a to the 11/12, b to the 10/12.
And then, finally, we could notice
that we have the same denominator on both,
which could be interpreted as the 12th index
on a to the 11th, b to the 10th.
However, that's a lot of work, so let's see if we can recognize
what happened as we did this problem.
You may notice that what we did to make it all possible
is we had to get a common denominator.
Those denominators came from the index,
so maybe instead of thinking of a common denominator,
we think about getting a common index.
And we can get a common index
by multiplying by the same thing on top and bottom,
or by multiplying...
the index... and the exponent...
by the same thing.
In other words,
without using rational exponents,
if I wanted an index of 12,
we could multiply the first index by 4
and the second by 3 to give us a 12th root of a to the...
multiplying the exponents also...
...8th, b to the 4th, and a to the 3rd, b to the 6th.
When we combine those together, we would have the 12th root
of a to the 11th, b to the 10th.
And in two steps,
we accomplished much quicker what we did before.
When there's numbers,
what we'll want to do is factor any numbers,
so we can use our exponent properties on those as well.
And as usual,
you always want to be sure your final answer is simplified.
Let's take a look at some examples
where we work through this process.
In this first problem,
we have an index of 4 and an index of 6.
Thinking about those as a denominator,
the common denominator would be 12.
This means we need to multiply
the first index by 3 and the second index by 2
to get a common index of 12 over the entire thing.
We'll also multiply our exponents by 3 and 2 as well,
giving us m to the 3; times 3, or 9th;
n to the 3; times 2, or 6th; p to the 3rd.
And now, multiplying by 2, m squared;
n to the 2; times 2, or 4; p to the 2; times 3, or 6.
Now, we can quickly combine our m's, n's, and p's together
by adding the exponents,
because the bases are multiplied.
Under the 12th root, we have m to the 9; plus 2, or 11;
n to the 6; plus 4, or 10; and p to the 3; plus 6, or 9.
We're left with the 12th root
of m to the 11th, n to the 10th, p to the 9th.
In Part 2 of this video, we'll look at a second example
that not only includes coefficients
that we need to factor,
but also the final answer would be reducible from here.