Divide and simplify. Assume that all variables are positive.
sqrt 64x^8y^9/sqrt8x^6y^8
To divide and simplify the expression, we will use the properties of square roots.
First, let's simplify the numerator and denominator separately before dividing.
In the numerator:
√(64x^8y^9)
We can simplify the square root by breaking down 64, x^8, and y^9 into their perfect square factors:
√(8^2 * (x^4)^2 * (y^4)^2 * y)
Using the properties of square roots, we can rewrite this as:
8 * x^4 * y^4 * √y
Now, let's simplify the denominator:
√(8x^6y^8)
Breaking down 8, x^6, and y^8 into their perfect square factors:
√(2^3 * (x^3)^2 * (y^4)^2)
Simplifying further:
2 * x^3 * y^4 * √2
Now, we can divide the numerator by the denominator:
(8 * x^4 * y^4 * √y) / (2 * x^3 * y^4 * √2)
Simplifying the fraction:
(4x) / √2
Therefore, the simplified expression is 4x/√2.
To divide and simplify the expression sqrt(64x^8y^9) / sqrt(8x^6y^8), we can start by simplifying each square root individually.
Step 1: Simplify the numerator.
sqrt(64x^8y^9) = sqrt(8^2 * x^4 * y^8 * y) = 8x^4y^4sqrt(y)
Step 2: Simplify the denominator.
sqrt(8x^6y^8) = sqrt(4 * 2 * x^6 * y^8) = 2x^3y^4sqrt(2)
Step 3: Divide the simplified numerator by the simplified denominator.
[8x^4y^4sqrt(y)] / [2x^3y^4sqrt(2)] = (8/2)(x^4/x^3)(y^4/y^4)(sqrt(y)/sqrt(2))
Step 4: Simplify the terms with the same base.
= 4x(y/√2)
Therefore, the simplified expression is 4xy√(y/2).