Combine the following rational expressions
1/2y-5-30/8y^3-125
Lcm:(2y-5)(4y^2+10y+125)
=1*4y^2+10y+25/(2y-5)(4y^2+10y+25)-30/(2y-5)(4y^2+10y+25)
=4y^2-10y+25-30/(2y-5)(4y^2+10y+25)
=(4y^2+10y-5)/(2y-5)(4y^2+10y+25) Please correct my answer and I know it is wrong please explain the steps completely.
1/(2y-5) - 30/(8y^3-125)
1/(2y-5) - 30/((2y)^3 - 5^3)
1/(2y-5) - 30/((2y-5)(4y^2+10y+25))
The LCM is just (8y^3-125), so that gives
((4y^2+10y+25)-30)/(8y^3-125) = (4y^2+10y-5)/(8y^3-125)
actually, your answer is correct, despite the typo
To combine the given rational expressions, you need to find the least common denominator (LCD) of the two expressions.
Given expressions:
1) (1/2y - 5)
2) (30/8y^3 - 125)
Step 1: Factorize the denominators:
1) Denominator of the first expression: (2y - 5)
2) Denominator of the second expression: (8y^3 - 125)
Step 2: Simplify the second expression:
The denominator of the second expression is a difference of cubes, which can be factored using the formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Thus, (8y^3 - 125) can be factored as: (2y - 5)(4y^2 + 10y + 25)
The LCD is the product of the factored denominators, which gives: (2y - 5)(4y^2 + 10y + 25)
Step 3: Write the two expressions with the LCD as the denominator:
1) First expression: (1/2y - 5) --> (1(4y^2 + 10y + 25))/(2y - 5)(4y^2 + 10y + 25)
2) Second expression: (30/8y^3 - 125) --> (30)/(2y - 5)(4y^2 + 10y + 25)
Step 4: Combine the two expressions:
(1(4y^2 + 10y + 25))/(2y - 5)(4y^2 + 10y + 25) - (30)/(2y - 5)(4y^2 + 10y + 25)
Step 5: Simplify the numerator and subtract the fractions:
The numerators do not have any common factors, so we can just subtract them: (4y^2 + 10y + 25 - 30)/(2y - 5)(4y^2 + 10y + 25)
Simplifying the numerator further:
(4y^2 + 10y - 5)/(2y - 5)(4y^2 + 10y + 25)
Thus, the combined rational expression is:
(4y^2 + 10y - 5)/(2y - 5)(4y^2 + 10y + 25)