Final answer:
The simplest form of √126xy⁵/32x³ is 3y²√(14y)/8x, achieved by factoring out square numbers and simplifying the radicals and exponents.
Step-by-step explanation:
The simplest form of √126xy&sup5;/32x³ is 3y²√(14y)/8x. To simplify the expression, start by breaking down the square root. The number 126 can be factored into 9 * 14, where 9 is a perfect square. This knowledge helps us simplify the square root part of the numerator. Combining this with the properties of exponents, which you can add when multiplying like bases, and using the fact that x² = √x, allows us to simplify the expression.
Since start equation 126 equals 9 multiplied by 14 end equation, take the square root of 9 (which is 3) out of the radical. In the exponent y&sup5, take y&sup4; (=y²²) out of the square root, so what remains inside the root is y. The denominator 32x³ simplifies to 16 * 2x³, where 16 is a perfect square.
Finally, we have √126xy&sup5;(equals 3y²√14y) over √32x³(equals 4x√2x). Simplifying further gives:
3y²√(14y)/8x