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Advanced Math / Equivalent expressions Difficulty: Hard

Which expression is equivalent to $\frac{42 a}{k} + 42 a k$ , where $k greater than 0$ ?

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Explanation

Choice D is correct. Two fractions can be added together when they have a common denominator. Since $k greater than 0$ , multiplying the second term in the given expression by $\frac{k}{k}$ yields $StartFraction left parenthesis 42 a k right parenthesis k Over k EndFraction$ , which is equivalent to $StartFraction 42 a k squared Over k EndFraction$ . Therefore, the expression $\frac{42 a}{k} + 42 a k$ can be written as $\frac{42 a}{k} + \frac{42 a k^{2}}{k}$ which is equivalent to $\frac{42 a + 42 a k^{2}}{k}$ . Since each term in the numerator of this expression has a factor of $42 a$ , the expression $\frac{42 a + 42 a k^{2}}{k}$ can be rewritten as $\frac{42 a (1) + 42 a (k^{2})}{k}$ , or $\frac{42 a (1 + k^{2})}{k}$ , which is equivalent to $\frac{42 a (k^{2} + 1)}{k}$ .

Choice A is incorrect. This expression is equivalent to $\frac{42 a}{k} + \frac{42 a}{k}$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This expression is equivalent to $\frac{42 a}{k} + 42 a$ .