Simplify the Complex Fractions
1. 8a / 3b^2 / 6ab /5
2. 42x^2 + 14x / x - 7 / 2 + 6x / 21 - 3x
3. 9 + 3 / x / x / 4 + 1 /12
4. (x^-1 + x^-2)divided by (9x^-2)
Ans. #1: 20 / 9
Ans. #2: 42x^2 + 6x + 4 / -2x + 14
Ans. #3: 144 / x / 3x + 1
I don't know how to do #4
I have trouble interpreting you run-on compound fractions. How about some parentheses so we know what's what?
(x^-1 + x^-2)/(9-x^2)
(1/x + 1/x^2)/(9-x^2)
(x+1)/(x^2 (9-x^2))
This doesn't simplify much.
1. (8a / 3b^2) / (6ab / 5)
2. (42x^2 + 14x) / (x - 7) / (2 + 6x / 21 - 3x)
3. (9 + 3 / x) / (x / 4 + 1 /12)
1. (8a)/(3b^2) * (5)/(6ab) = (40a)/(18ab^3) = 20/(9b^3)
2. (14x)(3x+1)/(x-7) * 3(7-x) / (2(3x+1))
= 42x(3x+1)(7-x) / 2(x-7)(3x+1)
= -21x
3. (9x+3)/x / (3x+1)/12
= 3(3x+1)/x * 12/(3x+1)
= 36/x
Thank you very much, Steve :)
To simplify complex fractions, we need to follow a few steps. Let's go through each of the given complex fractions and simplify them one by one.
1. 8a / 3b^2 / 6ab / 5:
To simplify this, we need to divide the dividend by the divisor. In this case, we can rewrite the expression as:
(8a / 3b^2) * (5 / 6ab)
We can cancel out common factors as follows:
(8a * 5) / (3b^2 * 6ab)
40a / (18ab^3)
We can further simplify this by cancelling out common factors:
40 / (18b^2)
Dividing both the numerator and denominator by the greatest common factor (2):
20 / (9b^2)
Therefore, the simplified form is 20 / 9b^2.
2. 42x^2 + 14x / x - 7 / 2 + 6x / 21 - 3x:
For this expression, we need to simplify it by applying the order of operations (PEMDAS/BODMAS). We should simplify within parentheses first, then perform multiplication and division from left to right, and finally addition and subtraction from left to right. Let's simplify this step by step:
Start by simplifying within parentheses:
= (42x^2 + 14x) / (x - 7) / (2 + 6x) / (21 - 3x)
= (42x^2 + 14x) / (x - 7) / (2 + 6x) / (21 - 3x)
Now, let's simplify the division by inverting the second fraction (2 + 6x) / (21 - 3x) and multiplying:
= (42x^2 + 14x) / (x - 7) * (21 - 3x) / (2 + 6x)
= [42x^2 + 14x] * [21 - 3x] / [(x - 7) * (2 + 6x)]
= (42x^2 * 21 + 42x^2 * (-3x) + 14x * 21 + 14x * (-3x)) / [(x - 7) * (2 + 6x)]
= (882x^2 - 126x^3 + 294x - 42x^2) / [(x - 7) * (2 + 6x)]
Now combine like terms:
= (-126x^3 + 840x^2 + 294x) / [(x - 7) * (2 + 6x)]
And that is the simplified form of the given complex fraction.
3. 9 + 3 / x / x / 4 + 1 / 12:
We'll follow the same process as mentioned above.
= 9 + 3 / x / x / 4 + 1 / 12
Now, let's simplify the division by inverting the second fraction (x / x) and (4 / 12) and multiplying:
= 9 + 3 * (4 / x) * (12 / 1)
= 9 + 3 * (4 * 12) / x
= 9 + 3 * 48 / x
= 9 + 144 / x
Therefore, the simplified form is 144 / x + 9.
4. (x^-1 + x^-2) divided by (9x^-2):
To simplify this complex fraction, combine the terms inside the parentheses:
= (1/x + 1/x^2) / (9/x^2)
Now, invert the second fraction and multiply:
= (1/x + 1/x^2) * (x^2/9)
Apply multiplication:
= (1 + x) / 9
Therefore, the simplified form is (1 + x) / 9.
I hope this clarifies how to simplify complex fractions!