Part 1Add. Simplify, if possible.StartFraction 6 Over y plus 6 EndFraction plus StartFraction 2 Over 7 y EndFraction plus StartFraction 36 Over y squared plus 6 y EndFraction6y+6+27y+36y2+6y Part 1The sum is [input]StartFraction 44 Over 7 y EndFraction44 7y. (Simplify your answer.)

Question

Part 1Add.​ Simplify, if possible.StartFraction 6 Over y plus 6 EndFraction plus StartFraction 2 Over 7 y EndFraction plus StartFraction 36 Over y squared plus 6 y EndFraction6y+6+27y+36y2+6y Part 1The sum is [input]StartFraction 44 Over 7 y EndFraction44 7y. ​(Simplify your​ answer.)

BlackTom AI · Accepted Answer

Find a common denominator: the least common denominator is 7y(y + 6) because y + 6, 7y, and y(y + 6) are the individual denominators.

1. Relevant knowledge points and concepts
- Adding rational expressions requires a common denominator.
- Factor denominators when possible (y^2 + 6y = y(y + 6)).
- Combine numerators over the common denominator and simplify by canceling common factors.
- Note domain restrictions: any value making an original denominator zero (y = 0 or y = −6) is excluded.

2. Step-by-step reasoning and conversion to common denominator
Start with: 6/(y + 6) + 2/(7y) + 36/(y^2 + 6y).
Factor the third denominator: y^2 + 6y = y(y + 6).
Common denominator: 7y(y + 6).

Convert each term:
- 6/(y + 6) = 6 * 7y / [7y(y + 6)] = 42y / [7y(y + 6)]
- 2/(7y) = 2 * (y + 6) / [7y(y + 6)] = (2y + 12) / [7y(y + 6)]
- 36/(y(y + 6)) = 36 * 7 / [7y(y + 6)] = 252 / [7y(y + 6)]

3. Add numerators (showing intermediate arithmetic)
Sum numerator: 42y + (2y + 12) + 252
Combine like terms: (42y + 2y) + (12 + 252) = 44y + 264

So the combined fraction is (44y + 264) / [7y(y + 6)].

4. Simplify and explain cancellation
Factor the numerator: 44y + 264 = 44(y + 6).
Thus the expression becomes 44(y + 6) / [7y(y + 6)].
Cancel the common factor (y + 6) (valid since y ≠ −6): result = 44 / (7y).

5. Why other answers would be incorrect
Common mistakes: failing to use the full LCD 7y(y + 6), adding numerators without converting to the LCD, arithmetic errors in combining constants (e.g., forgetting 252), or incorrectly canceling factors of y when they don't appear as factors in the whole numerator. Any such error leads to forms different from 44/(7y).

6. Conclusion and confirmation
After converting to the LCD, combining terms, factoring, and canceling, the simplified sum is 44/(7y), which matches the provided correct answer. (Domain: y ≠ 0, y ≠ −6.)

Part 1Add. Simplify, if possible.StartFraction 6 Over y plus 6 EndFraction plus StartFraction 2 Over 7 y EndFraction plus StartFraction 36 Over y squared plus 6 y EndFraction6y+6+27y+36y2+6y Part 1The sum is [input]StartFraction 44 Over 7 y EndFraction44 7y. (Simplify your answer.)多项填空题

类似问题

Part 1Use any method to simplify the complex fraction. StartStartFraction StartFraction k plus 5 Over 8 k EndFraction OverOver StartFraction 5 k minus 1 Over 2 k EndFraction EndEndFractionk+58k 5k−12k Part 1The fraction simplifies to [input]enter your response here .(Simplify your answer.)

Part 1Add. Simplify, if possible.StartFraction 6 Over y plus 6 EndFraction plus StartFraction 2 Over 7 y EndFraction plus StartFraction 36 Over y squared plus 6 y EndFraction6y+6+27y+36y2+6y Part 1The sum is [input]enter your response here . (Simplify your answer.)

What is the result when the fraction 𝑡 𝑡 + 2 − 1 + 𝑡 𝑡 − 3 is simplified as far as possible?

Given , calculate the quotient .

Subtract the rational expressions: 3y+2 y2−2y+1 − 2y+3 y2−2y+1 Simplify if possible.

Subtract the rational expressions: 13 9+p2 − 5 9+p2 Simplify if possible.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单