Question

simplify each expression completely.

1. $5\sqrt{7}-6\sqrt{7}$
2. $\sqrt{5}+\sqrt{20}$
3. $3\sqrt{2}(\sqrt{5}+6)$
4. $4\sqrt{6}(\sqrt{12}-\sqrt{2})$

solve each radical equation for the variable.

1. $\sqrt{x}+9=16$
2. $3=\sqrt{2m-1}$

Explanation:

Problem 16

Step1: Combine like radical terms

$5\sqrt{7} - 6\sqrt{7} = (5-6)\sqrt{7}$

Step2: Compute coefficient difference

$(5-6)\sqrt{7} = -1\sqrt{7} = -\sqrt{7}$

Problem 17

Step1: Simplify $\sqrt{20}$

$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

Step2: Add like radical terms

$\sqrt{5} + 2\sqrt{5} = (1+2)\sqrt{5} = 3\sqrt{5}$

Problem 18

Step1: Distribute $3\sqrt{2}$

$3\sqrt{2} \times \sqrt{5} + 3\sqrt{2} \times 6$

Step2: Simplify each term

$3\sqrt{10} + 18\sqrt{2}$

Problem 19

Step1: Simplify $\sqrt{12}$

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

Step2: Distribute $4\sqrt{6}$

$4\sqrt{6} \times 2\sqrt{3} - 4\sqrt{6} \times \sqrt{2}$

Step3: Multiply radicals and coefficients

$8\sqrt{18} - 4\sqrt{12}$

Step4: Simplify resulting radicals

$8\sqrt{9 \times 2} - 4\sqrt{4 \times 3} = 8 \times 3\sqrt{2} - 4 \times 2\sqrt{3} = 24\sqrt{2} - 8\sqrt{3}$

Problem 20

Step1: Isolate the radical term

$\sqrt{x} = 16 - 9$

Step2: Simplify right-hand side

$\sqrt{x} = 7$

Step3: Square both sides

$x = 7^2 = 49$

Problem 21

Step1: Square both sides

$3^2 = 2m - 1$

Step2: Simplify left-hand side

$9 = 2m - 1$

Step3: Isolate the variable term

$2m = 9 + 1$

Step4: Solve for $m$

$2m = 10 \implies m = \frac{10}{2} = 5$

Answer:

1. $-\sqrt{7}$
2. $3\sqrt{5}$
3. $3\sqrt{10} + 18\sqrt{2}$
4. $24\sqrt{2} - 8\sqrt{3}$
5. $x=49$
6. $m=5$