Question 225·Hard·Advanced Math

Since $k>0$, choose $k=2$ rather than $1$, where a subtraction could become zero and hide a sign error. Type $k_1=2$, then type the original expression with $k_1$ in place of $k$. Desmos prints $0.4375$.

Type the first two choices below the original, still using $k_1$. Desmos prints $-0.4375$ and $0.5625$, respectively. Both differ from $0.4375$, so neither expression is equivalent to the original. Algebra can settle the other forms without relying on one test.

Because $k>0$, $k^{1/2}$ is nonzero. Multiply the top and bottom of the first fraction by it:

Add exponents when multiplying powers of $k$:

Divide top and bottom by $2$:

The negative exponent $k^{-1}$ means $1/k$. Multiply the top and bottom of the second fraction by $k$ to clear it:

Add exponents on matching powers of $k$:

Use $k^0=1$, since $k>0$:

Multiply the numerators together and the denominators together. Powers of the same base add when multiplied, so $k\cdot k^3=k^4$:

Reversing a subtraction changes its sign. Reverse both numerator factors, not only one:

The two negative signs multiply to a positive:

So this expression is equivalent to the original for every $k>0$. Choice D.