What is the simplest form of the quotient?
^4√810/^4√2
To find the simplest form of the quotient ^4√810/^4√2, we need to simplify each radical separately and then divide them.
Let's start with ^4√810. In this case, the index is 4, which means we need to find the fourth root of 810. To simplify a fourth root, we need to break down the number 810 into its factor pairs.
The prime factorization of 810 is 2 * 3 * 3 * 3 * 3 * 3 * 5.
Now we group these factors in sets of four, since we are taking the fourth root:
(2 * 3 * 3 * 3) * (3 * 3 * 3 * 5)
Simplifying this, we get:
= 18 * 27
= 486
So, ^4√810 simplifies to √486.
Now let's simplify the other radical, ^4√2. Since 2 is a prime number, its fourth root is still ^4√2.
Now we can divide the simplified radicals:
√486 / ^4√2
Simplifying the division of radicals involves multiplying the numerator and denominator by a rationalizing factor that eliminates the fourth root in the denominator.
In this case, we can multiply both the numerator and denominator by (√2)^3. This will make the denominator become ^4√(2^3 * 2) = ^4√16 = 2.
The simplified form of √486 / ^4√2 is therefore (√486 * (√2)^3) / (2 * (√2)^3) which simplifies to (√972) / (2√2).
Finally, (√972) is equal to √(324 * 3), which simplifies to 18√3.
Therefore, the simplest form of the quotient ^4√810/^4√2 is 18√3 / 2√2.
I assume it is this :
(810)^.25 / (2)^.25
or=(405)^.25
=(81*5)^.25
=3*(5)^.25