15sqrt8x^15/5sqrt2x^4
remember that sqrt(a)/sqrt(b) = sqrt(a/b)
simplify the inside.
also notice that sqrt(x^11) = sqrt(x^10) * sqrt(x)
= x^5 * sqrt(x)
To simplify the expression (15sqrt8x^15)/(5sqrt2x^4), we can follow these steps:
Step 1: Apply the quotient rule for square roots: sqrt(a) / sqrt(b) = sqrt(a/b).
(15sqrt8x^15) / (5sqrt2x^4) = (15/5) * (sqrt(8x^15) / sqrt(2x^4))
Step 2: Simplify the inside of the square roots.
Let's simplify the numerator first. Recall that sqrt(a*b) = sqrt(a) * sqrt(b).
sqrt(8x^15) = sqrt(8) * sqrt(x^15)
Next, we need to simplify these individual square roots further.
sqrt(8) = sqrt(4 * 2) = sqrt(4) * sqrt(2) = 2 * sqrt(2)
For the second square root, we can use the property sqrt(x^m) = x^(m/2), where x is non-negative.
sqrt(x^15) = x^(15/2)
Putting it all together:
(15/5) * (sqrt(8x^15) / sqrt(2x^4)) = 3 * (2 * sqrt(2) * x^(15/2)) / (sqrt(2x^4))
Step 3: Simplify further if possible.
We can simplify by canceling out the square root of 2 on the numerator and denominator:
= 3 * (2 * x^(15/2)) / (sqrt(x^4))
= 3 * (2 * x^(15/2)) / (x^2)
Finally, we can simplify the expression further:
= 6 * x^(15/2 - 2)
= 6 * x^(15/2 - 4/2)
= 6 * x^(11/2)
= 6 * sqrt(x^11) * x
= 6x * sqrt(x^11)
So, the simplified expression is 6x * sqrt(x^11).