1) To simplify the expression (6x^3-16x^2+11x-5)/(3x-2), you can use polynomial long division or synthetic division. Here's how to do it using polynomial long division:
- Write the dividend (6x^3-16x^2+11x-5) as the numerator and the divisor (3x-2) as the denominator.
- Divide the first term of the numerator (6x^3) by the first term of the denominator (3x). The result is 2x^2.
- Multiply the divisor (3x-2) by the result (2x^2) and subtract it from the numerator: (6x^3-16x^2+11x-5) - (2x^2 * (3x-2)) = 2x^2(3x-2)-16x^2+11x-5.
- Repeat the process with the new expression: divide the first term (2x^2) by the first term of the denominator (3x). The result is (2/3)x. Multiply the divisor (3x-2) by the result [(2/3)x] and subtract it from the expression.
- Continue with the process until you have no more terms left to divide.
- The resulting quotient is 2x^2-4x+1, and the remainder is -3.
- Therefore, the simplified expression is (2x^2-4x+1)-3/(3x-2), which can be further simplified as 2x^2-4x-2/(3x-2) by combining like terms.
2) To factor the expression 27x^3-1 completely, you can use the difference of cubes formula, which states that a^3-b^3 = (a-b)(a^2+ab+b^2).
In this case, the expression can be rewritten as (3x)^3 - 1^3. Applying the difference of cubes formula, we get:
27x^3-1 = (3x-1)(9x^2+3x+1)
Therefore, the expression 27x^3-1 can be factored completely as (3x-1)(9x^2+3x+1).
3) To simplify the expression (x^2-3x-28)/(x^2-9x+14), you can factor both the numerator and the denominator, if possible, and cancel any common factors.
In this case, the numerator and the denominator can be factored as:
Numerator: x^2-3x-28 = (x-7)(x+4)
Denominator: x^2-9x+14 = (x-2)(x-7)
Now, we can cancel out the common factor (x-7) from both the numerator and the denominator:
(x^2-3x-28)/(x^2-9x+14) = (x-7)(x+4)/(x-2)(x-7)
Canceling out the common factor (x-7), we get:
(x-7)(x+4)/(x-2)(x-7) = (x+4)/(x-2)
Therefore, the simplified expression is (x+4)/(x-2).
4) To simplify the square root of 64n^6w^4, you can break down the number inside the square root sign into its prime factors and then take out any perfect squares.
64 can be written as 8 * 8 = 2^6
n^6 can be written as n^3 * n^3
w^4 can be written as w^2 * w^2
So, the expression can be rewritten as:
√(64n^6w^4) = √(2^6 * n^3 * n^3 * w^2 * w^2)
Taking out the perfect squares, we have:
= √(2^6) * √(n^3 * n^3) * √(w^2 * w^2)
Simplifying further:
= 8 * n^3 * w^2
Therefore, the simplified expression is 8n^3w^2.
5) The question is a bit unclear, but assuming you're looking for the square root of 625x^5 with a 3 on top of the square root sign, it can be simplified as follows:
∛(625x^5) = ∛(5^4 * x^5)
Since 5^4 is a perfect cube, we can take it outside the cube root sign:
∛(5^4 * x^5) = 5 * ∛(x^5)
The cube root of x^5 can be simplified as:
∛(x^5) = ∛(x^3 * x^2) = x^2 * ∛(x)
Therefore, the simplified expression is 5x^2∛(x).