### How Many Solutions?

Find all the solutions to the this equation.

### Quartics

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

### Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

# Pitchfork

##### Age 16 to 18Challenge Level

Consider the symmetry of the graph. Write the equation in the form

$$x^{1/x}=y^{1/y}$$

and consider the graph of $y=x^{1/x}$ from the question Exponential Trends.