Let \( \mathbf{u}=\langle 9,-8\rangle, \mathbf{v}=\langle-1,7\rangle \), and \( \mathbf{w}=\langle 0,-8\rangle \). Carry out the following computation. Find \( |\mathbf{u}+\mathbf{v}+\mathbf{w}| \). \( |\mathbf{u}+\mathbf{v}+\mathbf{w}|=\square \) (Type an exact answer, using radicals as needed.)

To find the magnitude of the vector \( \mathbf{u} + \mathbf{v} + \mathbf{w} \), follow these steps: 1. **Add the vectors component-wise:** \[ \mathbf{u} + \mathbf{v} + \mathbf{w} = \langle 9, -8 \rangle + \langle -1, 7 \rangle + \langle 0, -8 \rangle \] \[ = \langle 9 + (-1) + 0, -8 + 7 + (-8) \rangle \] \[ = \langle 8, -9 \rangle \] 2. **Calculate the magnitude of the resulting vector:** The magnitude \( |\mathbf{a}| \) of a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) is given by: \[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2} \] Applying this to \( \mathbf{u} + \mathbf{v} + \mathbf{w} = \langle 8, -9 \rangle \): \[ |\mathbf{u} + \mathbf{v} + \mathbf{w}| = \sqrt{8^2 + (-9)^2} = \sqrt{64 + 81} = \sqrt{145} \] So, the exact magnitude is: \[ |\mathbf{u} + \mathbf{v} + \mathbf{w}| = \sqrt{145} \] **Answer:** \(\sqrt{145}\)