Let's solve this radical equation step by step. We have negative three times the quantity x minus two to the three-halves power plus seven equals negative seventeen. First, we'll isolate the radical term by subtracting seven from both sides, giving us negative three times the quantity x minus two to the three-halves power equals negative twenty-four. Then we divide both sides by negative three to get the quantity x minus two to the three-halves power equals eight.
Now we have the quantity x minus two to the three-halves power equals eight. To eliminate the fractional exponent, we raise both sides to the reciprocal power, which is two-thirds. This gives us x minus two equals eight to the two-thirds power. Let's calculate eight to the two-thirds. We can rewrite this as eight to the one-third squared, which equals two squared, which equals four. So we have x minus two equals four.
Now we have x minus two equals four. To solve for x, we add two to both sides, giving us x equals six. Let's verify this answer by substituting x equals six back into the original equation. We get negative three times the quantity six minus two to the three-halves power plus seven. This simplifies to negative three times four to the three-halves power plus seven. Since four to the three-halves equals eight, we have negative three times eight plus seven, which equals negative twenty-four plus seven, which equals negative seventeen. This matches our original right-hand side, so our solution is correct.
Let's visualize the function y equals the quantity x minus two to the three-halves power. This graph shows how the function behaves. Notice that the domain starts at x equals two, since we need x minus two to be non-negative for the fractional exponent to be real. At our solution x equals six, we can see that y equals eight, which matches our equation. The red dot shows the point where x equals six and y equals eight, confirming our solution graphically.