1)Simplify:(6x^3-16x^2+11x-5)/(3x-2)
answer= 2x^2-4x+1-3over3x-2
2)Factor 27x^3-1 completley.
answer= (3x-1)^3 if this isnt correct,im not sure how to solve it b/c this makes the most sense to me.
3)Simplify: x^2-3x-28 over x^2-9x+14,assume that the denominator is not equal to 0.
answer= x+4 over x-2
4)Simplify: square root of 64n^6w^4
answer= 8n^3w^2
5)Simplify: (3 is on the point of the square root sign,not on top of 625) square root of 625x^5
answer= 25x^2
3 is correct.
4 is correct.
If you are trying to get cuberoot 625x^5
then rewrite it as
cubroot (5^4 * x^5)
cuberoot(5*5^3*x^3*x^3)
then pull the perfect cubes out.
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).
I apologize for any confusion caused. Let's clarify and provide the correct steps for the questions.
1) Simplify: (6x^3-16x^2+11x-5)/(3x-2)
To simplify this expression, you can use long division or synthetic division.
Using long division:
Divide 6x^3 by 3x: 2x^2
Multiply (3x-2) by 2x^2: 6x^3 - 4x^2
Subtract this from the original expression: (-16x^2 + 11x - 5) - (-4x^2) = -16x^2 + 11x - 5 + 4x^2 = -12x^2 + 11x - 5
Now divide -12x^2 by 3x: -4x
Multiply (3x-2) by -4x: -12x^2 + 8x
Subtract this from the previous result: (11x - 5) - (8x) = 11x - 5 - 8x = 3x - 5
Finally, divide 3x by 3x: 1
Multiply (3x-2) by 1: 3x - 2
Subtract this from the previous result: (3x - 5) - (3x - 2) = 3x - 5 - 3x + 2 = -3
Therefore, the simplified expression is 2x^2 - 4x + 1 - 3/(3x - 2).
2) Factor 27x^3 - 1 completely.
This expression can be factored using the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
In this case, a = 3x and b = 1.
Applying the formula, we have:
27x^3 - 1 = (3x - 1)(9x^2 + 3x + 1)
Hence, the expression 27x^3 - 1 can be factored as (3x - 1)(9x^2 + 3x + 1).
3) Simplify: (x^2 - 3x - 28)/(x^2 - 9x + 14), assuming the denominator is not equal to 0.
To simplify this rational expression, you can factor the numerator and denominator and then cancel any common factors.
Factoring the numerator, we have:
x^2 - 3x - 28 = (x - 7)(x + 4)
Factoring the denominator, we have:
x^2 - 9x + 14 = (x - 2)(x - 7)
Now we can simplify the expression:
(x^2 - 3x - 28)/(x^2 - 9x + 14) = ((x - 7)(x + 4))/((x - 2)(x - 7))
The common factor (x - 7) cancels out, leaving:
(x + 4)/(x - 2)
Hence, the simplified expression is (x + 4)/(x - 2).
4) Simplify: √(64n^6w^4)
To simplify this expression, you can take the square root of each factor individually, using the rule √(ab) = √a * √b.
√(64n^6w^4) = √(8^2 * (n^2)^3 * (w^2)^2)
Taking the square root of each factor, we have:
√(8^2) * √((n^2)^3) * √((w^2)^2)
Simplifying further:
8 * (n^2)^3/2 * (w^2)^1
Since (a^m)^n = a^(m*n), we can simplify the expression to:
8n^3w^2
Therefore, the simplified expression is 8n^3w^2.
5) Simplify: √(625x^5)
To simplify this expression, you can apply the property √(a^b) = a^(b/2).
√(625x^5) = (625x^5)^(1/2)
Using the property, we have:
625^(1/2) * (x^5)^(1/2)
Simplifying further:
25 * (x^2)^(1/2)
Since (a^m)^n = a^(m*n), we can simplify the expression to:
25x^2
Therefore, the simplified expression is 25x^2.