Final answer:
To solve √(18x−9)=x⁵, we square both sides resulting in a quintic equation, which is complex and may require numerical solutions or graphing methods for real solutions.
Step-by-step explanation:
To solve the radical equation √(18x−9)=x⁵, we start by squaring both sides to eliminate the square root. This gives us 18x - 9 = x²⁵. Now, let's simplify the equation further by rewriting x⁵ as (x²)²·x. Applying this, we have 18x - 9 = (x²)²·x. At this point, we recognize that we are dealing with a quintic equation, which generally does not have a straightforward solution. However, due to its specific form, we may attempt to find solutions through substitution, factoring, or numerical methods if necessary.
We can also examine for any potential rational roots using the Rational Root Theorem, yet this polynomial does not lend itself easily to factoring methods. The complexity of quintic equations usually requires numerical methods such as Newton's method for more accurate solutions, but it is beyond the scope of a typical high school curriculum to delve into such methods.
If we are asked only to find real solutions, we might graph both sides of the original equation as functions of x and look for intersections, or use a graphing calculator or computer software to approximate the solutions.