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Deep roots allow you to be chopped down, blown around, and burned up, yet you will grow, bloom, and produce fruit again. It is critical that you prune "diseased, damaged, dead, non-productive, structurally unsound, or unwanted," branches/areas of your life. This allows you to grow deep roots.
Preoccupation with appearances and a corresponding neglect of the out-of-sight root of the true spiritual life are prophetic signs which go unheeded. Tozer
Equally as important as deep roots is what you are rooted in. What you are rooted in feeds/motivates and moves/shakes you. If you are unsure what you are rooted in consider what shakes you. For example, if anger wells up in your heart and you allow it to move you then you likely have a root in anger.
If you are deeply rooted in love (as Paul encourages us to be in Ephesians 3:14-19) then you will not be shaken or moved by the cares of this world but will be filled with "all the fullness of God."