I need help simplifying the following problems.
1) square root 16x^2y^4
2) 8y^3+27 / 2xy-10y+3x-15
Sure! I can help you simplify these problems.
1) To simplify the expression √(16x^2y^4), we start by factoring the number under the square root sign. The square root of 16 is 4, and we can write it as the square root of 4 times the square root of x^2 times the square root of y^4.
So, the expression becomes 4√(x^2y^4).
Next, we apply the property of square roots that states √(ab) = √a * √b. We can rewrite the expression as 4 * √(x^2) * √(y^4).
Since the square root of x^2 is x (the square root "undoes" the square), and the square root of y^4 is y^2 (the square root "undoes" the exponent 2), we can simplify our expression further.
The final simplified expression is 4xy^2.
2) To simplify the expression (8y^3 + 27) / (2xy - 10y + 3x - 15), we start by factoring the numerator and the denominator.
The numerator, 8y^3 + 27, is already fully factored.
The denominator, 2xy - 10y + 3x - 15, can be factored by grouping. Rearranging the terms, we get (2xy + 3x) + (-10y - 15). Then, applying the distributive property, we can factor out common terms in each group:
2x(y + (3/2)) - 5(2y + 3).
Now, the expression becomes (8y^3 + 27) / [2x(y + 3/2) - 5(2y + 3)].
We can simplify further by rewriting the numerator as 2^3 * y^3 + 3^3:
(2^3 * y^3 + 3^3) / [2x(y + 3/2) - 5(2y + 3)].
This becomes (2y + 3)(4y^2 - 6y + 9) / [2xy + 3x - 10y - 15].
So, the simplified expression is (2y + 3)(4y^2 - 6y + 9) / [2xy + 3x - 10y - 15].