solve it---Question Number: 23 Question Stem: Which expression is equivalent to (42a)/k + 42ak, where k > 0 ? Options: A) (84a)/k B) (84ak^2)/k C) (42a(k + 1))/k D) (42a(k^2 + 1))/k

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Let's solve Question 23. We need to find which expression is equivalent to 42a divided by k, plus 42a times k, where k is greater than zero. To combine these terms, we need a common denominator. The first term already has a denominator of k. For the second term, 42a times k, we can rewrite it with denominator k by multiplying both numerator and denominator by k. This gives us 42a times k squared over k. Now we can add the fractions: 42a over k plus 42a k squared over k equals 42a plus 42a k squared, all over k. We can factor out 42a from the numerator to get 42a times 1 plus k squared, all over k. This can be written as 42a times k squared plus 1, all over k. Looking at our options, this matches answer choice D. Let's work through this step by step. We start with the expression 42a over k plus 42a times k. First, we need to find a common denominator. The first term already has denominator k. For the second term, we multiply both numerator and denominator by k, giving us 42a k squared over k. Now we can add the fractions with the common denominator k. This gives us 42a plus 42a k squared, all over k. Next, we factor out the common term 42a from the numerator, resulting in 42a times 1 plus k squared, all over k. Finally, we can rearrange the terms inside the parenthesis to get 42a times k squared plus 1, all over k. Comparing with the given options, we can see that this matches option D. Now let's analyze why the other answer choices are incorrect. Option A suggests 84a over k. This is wrong because it simply adds the coefficients without properly accounting for the different terms. We can't just add 42a over k and 42ak to get 84a over k. Option B gives us 84a k squared over k. This is incorrect because it doubles both terms and applies k squared to both, which doesn't follow the rules of algebra. Option C suggests 42a times k plus 1, all over k. This is close, but it incorrectly adds k instead of k squared. The second term is 42a times k, which when converted to the common denominator becomes 42a k squared over k, not 42a k over k. Finally, Option D, which is 42a times k squared plus 1, all over k, is the correct answer as we've shown through our step-by-step solution. Let's summarize the key takeaways from this problem. When adding algebraic fractions, the first step is to find a common denominator. Then, convert all terms to have the same denominator. Next, factor out any common terms in the numerator to simplify the expression. Always make sure to simplify the expression to its final form. And finally, verify your answer by checking all the given options. In this problem, we started with the expression 42a over k plus 42a times k, and through our algebraic manipulations, we arrived at the equivalent expression 42a times k squared plus 1, all over k, which corresponds to option D. Let's summarize what we've learned from Question 23. We were asked to find an expression equivalent to 42a over k plus 42a times k, where k is greater than zero. We approached this problem systematically. First, we identified that we needed a common denominator, which is k. Then, we converted the second term to a fraction with denominator k by multiplying both numerator and denominator by k. Next, we added the fractions with the common denominator. After that, we factored out the common term 42a from the numerator. Finally, we rearranged the terms in the numerator to get our answer. The correct answer is option D: 42a times k squared plus 1, all over k. This problem demonstrates the importance of using algebraic techniques to simplify expressions and combine like terms.