To simplify the expression \( 7\sqrt{150} + 6\sqrt{6} \), we start by simplifying \( \sqrt{150} \).
The number \( 150 \) can be factored as: \[ 150 = 25 \times 6 \] So, we can write: \[ \sqrt{150} = \sqrt{25 \times 6} = \sqrt{25} \times \sqrt{6} = 5\sqrt{6} \]
Now, we substitute \( \sqrt{150} \) back into the original expression: \[ 7\sqrt{150} = 7 \times 5\sqrt{6} = 35\sqrt{6} \]
Now, we can rewrite the entire expression: \[ 7\sqrt{150} + 6\sqrt{6} = 35\sqrt{6} + 6\sqrt{6} \]
We can now combine the like terms: \[ 35\sqrt{6} + 6\sqrt{6} = (35 + 6)\sqrt{6} = 41\sqrt{6} \]
Thus, the simplified form of the expression \( 7\sqrt{150} + 6\sqrt{6} \) is: \[ \boxed{41\sqrt{6}} \]