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.)

Question

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.)

BlackTom AI · Accepted Answer

Dividing by a fraction is equivalent to multiplying by its reciprocal, so rewrite the complex fraction as a product.

Start with the original expression:
((k + 5) / (8k)) ÷ ((5k − 1) / (2k))

Replace division by multiplication by the reciprocal of the second fraction:
((k + 5) / (8k)) * ((2k) / (5k − 1))

Multiply numerators and denominators:
Numerator = (k + 5) * (2k)
Denominator = (8k) * (5k − 1)

Write that explicitly:
((k + 5)(2k)) / ((8k)(5k − 1))

Cancel common factors. There is a factor k in both the numerator and denominator: (2k) in the numerator and (8k) in the denominator. Cancel k:
((k + 5) * 2) / (8 * (5k − 1))

Now simplify the numeric factor 2/8 = 1/4:
(k + 5) / (4(5k − 1))

State domain restrictions (so simplification is valid): the original expression is undefined when any denominator is zero, so k ≠ 0 and 5k − 1 ≠ 0 ⇒ k ≠ 1/5.

Why this is correct: the reciprocal step is the standard algebraic operation for dividing rational expressions; cancellation of the common multiplicative factor k is valid because we exclude k = 0; simplifying 2/8 to 1/4 produces the final reduced rational expression with no common factors between numerator and denominator.

Common incorrect approaches: canceling parts of sums (for example canceling k in k + 5 with a factor in the denominator) is invalid because cancellation only applies to common factors, not terms in a sum. Another error is forgetting to take the reciprocal of the second fraction or failing to cancel the k, which yields a non-simplified or wrong result.

Final answer (simplified): (k + 5) / (4(5k − 1)), which matches the provided correct answer.

类似问题

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.)

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.

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