Question

lt#2 i can evaluate square roots and simplify radical expressions

1. \\(\frac{3 - \sqrt{10}}{12}\\)

evaluate and simplify the expression when \\(x = -4\\) and \\(z = 3\\)

1. \\(\sqrt{x^2 z + x z^2}\\)

1. simplify the following expression: \\(3\sqrt{32} - 5\sqrt{2}\\)

Explanation:

Step1: Simplify $\sqrt{10}$ (no further simplification)

$\frac{3-\sqrt{10}}{12}$ is already in simplest form, since $\sqrt{10}$ cannot be reduced to a rational number or simplified radical with integer factors other than 1 and 10.

Step2: Substitute $x=-4,z=3$ into $\sqrt{x^2 z + xz^2}$

First calculate inside the radical:
$x^2 z + xz^2 = (-4)^2 \cdot 3 + (-4) \cdot 3^2 = 16 \cdot 3 + (-4) \cdot 9 = 48 - 36 = 12$
Then take the square root: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$

Step3: Simplify $3\sqrt{32} - 5\sqrt{2}$

Rewrite $\sqrt{32}$ as $\sqrt{16 \cdot 2} = 4\sqrt{2}$. Substitute back:
$3 \cdot 4\sqrt{2} - 5\sqrt{2} = 12\sqrt{2} - 5\sqrt{2}$
Combine like radicals:
$12\sqrt{2} - 5\sqrt{2} = (12-5)\sqrt{2} = 7\sqrt{2}$

Answer:

1. $\frac{3-\sqrt{10}}{12}$
2. $2\sqrt{3}$
3. $7\sqrt{2}$