loodi respluer los cadraos pettectos \( 400 p^{10} z^{8}-40 p^{5} z^{4}+1 \)

It seems like you're asking for help with the expression \( 400 p^{10} z^{8} - 40 p^{5} z^{4} + 1 \). This expression can be analyzed or factored if possible. To factor the expression, we can look for a pattern or use substitution. Notice that the expression resembles a quadratic in terms of \( p^5 z^4 \). Let's rewrite it: Let \( x = p^5 z^4 \). Then, we can express the original polynomial as: \[ 400 x^2 - 40 x + 1 \] Now, we can factor this quadratic expression. To do this, we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 400 \), \( b = -40 \), and \( c = 1 \). Calculating the discriminant: \[ b^2 - 4ac = (-40)^2 - 4 \cdot 400 \cdot 1 = 1600 - 1600 = 0 \] Since the discriminant is zero, there is one repeated root: \[ x = \frac{40}{2 \cdot 400} = \frac{40}{800} = \frac{1}{20} \] Thus, the quadratic can be factored as: \[ 400 \left( x - \frac{1}{20} \right)^2 \] Substituting back \( x = p^5 z^4 \): \[ 400 \left( p^5 z^4 - \frac{1}{20} \right)^2 \] Finally, we can express the original polynomial as: \[ 400 \left( p^5 z^4 - \frac{1}{20} \right)^2 \] This is the factored form of the expression \( 400 p^{10} z^{8} - 40 p^{5} z^{4} + 1 \).