Adding and Subtracting Radicals

Show your work for each problem.

1.
Which expression shows like radicals?
1. a) $3\sqrt{5}$ and $7\sqrt{5}$
2. b) $\sqrt{8}$ and $\sqrt{9}$
3. c) $4\sqrt{2}$ and $4\sqrt{3}$
4. d) $2\sqrt{3}$ and $2\sqrt{7}$
Answer: $3\sqrt{5}$ and $7\sqrt{5}$
$3\sqrt{5}$ and $7\sqrt{5}$ are like radicals because both have $\sqrt{5}$ (same radicand). The other pairs have different radicands.
2.
Simplify $5\sqrt{3} + 2\sqrt{3}$ and write the result as $\square\sqrt{3}$ . What number goes in the box?
Answer: 7
$5\sqrt{3} + 2\sqrt{3} = (5+2)\sqrt{3} = 7\sqrt{3}$ . Just add the coefficients and keep the radical.
3.
Simplify $9\sqrt{7} - 4\sqrt{7}$ and write the result as $\square\sqrt{7}$ . What number goes in the box?
Answer: 5
$9\sqrt{7} - 4\sqrt{7} = (9-4)\sqrt{7} = 5\sqrt{7}$ . Just subtract the coefficients and keep the radical.
4.
What is $\sqrt{5} + 4\sqrt{5}$ ?
1. a) $\sqrt{25}$
2. b) $5\sqrt{5}$
3. c) $4\sqrt{10}$
4. d) $4\sqrt{5}$
Answer: $5\sqrt{5}$
$\sqrt{5} + 4\sqrt{5} = 1\sqrt{5} + 4\sqrt{5} = (1+4)\sqrt{5} = 5\sqrt{5}$ . Don't forget the implicit coefficient of 1!
5.
Simplify: $6\sqrt{2} + 3\sqrt{2} - 4\sqrt{2}$
Answer: $5\sqrt{2}$
1. Are all terms like radicals? yes
2. Combine the coefficients: 6 + 3 - 4 = ? 5
3. $6\sqrt{2} + 3\sqrt{2} - 4\sqrt{2} = 5\sqrt{\square}$ . What number goes in the box? 2
6.
What is $\sqrt{12} + \sqrt{3}$ ?
1. a) $2\sqrt{3}$
2. b) $3\sqrt{3}$
3. c) $\sqrt{36}$
4. d) $\sqrt{15}$
Answer: $3\sqrt{3}$
$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ . So $\sqrt{12} + \sqrt{3} = 2\sqrt{3} + 1\sqrt{3} = 3\sqrt{3}$ .
7.
Simplify: $\sqrt{20} + \sqrt{45}$
Answer: $5\sqrt{5}$
1. Simplify: $\sqrt{20} = \square\sqrt{5}$ . What number goes in the box? 2
2. Simplify: $\sqrt{45} = \square\sqrt{5}$ . What number goes in the box? 3
3. Now add the like radicals: $2\sqrt{5} + 3\sqrt{5} = \square\sqrt{5}$ . What number goes in the box? 5
8.
Simplify $\sqrt{75} - \sqrt{27}$ and write the result as $\square\sqrt{3}$ . What number goes in the box?
Answer: 2
$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$ and $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$ . So $5\sqrt{3} - 3\sqrt{3} = 2\sqrt{3}$ .
9.
Which expression cannot be simplified further?
1. a) $\sqrt{8} + \sqrt{2}$
2. b) $\sqrt{12} + \sqrt{27}$
3. c) $3\sqrt{5} + \sqrt{5}$
4. d) $2\sqrt{3} + 5\sqrt{7}$
Answer: $2\sqrt{3} + 5\sqrt{7}$
$2\sqrt{3} + 5\sqrt{7}$ has different radicands (3 and 7, both prime) so they cannot be combined. The others can all be simplified: $\sqrt{8} + \sqrt{2} = 3\sqrt{2}$ , $3\sqrt{5} + \sqrt{5} = 4\sqrt{5}$ , $\sqrt{12} + \sqrt{27} = 5\sqrt{3}$ .
10.
Simplify: $\sqrt{50} + \sqrt{8} - \sqrt{18}$
Answer: $4\sqrt{2}$
1. Simplify: $\sqrt{50} = \square\sqrt{2}$ . What number goes in the box? 5
2. Simplify: $\sqrt{8} = \square\sqrt{2}$ . What number goes in the box? 2
3. Simplify: $\sqrt{18} = \square\sqrt{2}$ . What number goes in the box? 3
4. Combine: $5\sqrt{2} + 2\sqrt{2} - 3\sqrt{2} = \square\sqrt{2}$ . What number goes in the box? 4
11.
Simplify $3\sqrt{5} + 2\sqrt{3} + \sqrt{5} - 4\sqrt{3}$ and write the result as $a\sqrt{5} + b\sqrt{3}$ . What is $b$ ?
Answer: -2
Group like radicals: $(3\sqrt{5} + \sqrt{5}) + (2\sqrt{3} - 4\sqrt{3}) = 4\sqrt{5} + (-2\sqrt{3}) = 4\sqrt{5} - 2\sqrt{3}$ . So $b = -2$ .
12.
Simplify: $\sqrt{72} - \sqrt{32} + \sqrt{98}$
Answer: $9\sqrt{2}$
1. Simplify: $\sqrt{72} = \square\sqrt{2}$ . What number goes in the box? 6
2. Simplify: $\sqrt{32} = \square\sqrt{2}$ . What number goes in the box? 4
3. Simplify: $\sqrt{98} = \square\sqrt{2}$ . What number goes in the box? 7
4. Combine: $6\sqrt{2} - 4\sqrt{2} + 7\sqrt{2} = \square\sqrt{2}$ . What number goes in the box? 9
13.
A triangle has sides $\sqrt{44}$ m, $\sqrt{99}$ m, and $\sqrt{176}$ m. What is its perimeter?
1. a) $9\sqrt{11}$ m
2. b) $7\sqrt{11}$ m
3. c) $6\sqrt{11}$ m
4. d) $\sqrt{319}$ m
Answer: $9\sqrt{11}$ m
$\sqrt{44} = 2\sqrt{11}$ , $\sqrt{99} = 3\sqrt{11}$ , $\sqrt{176} = 4\sqrt{11}$ . Perimeter = $2\sqrt{11} + 3\sqrt{11} + 4\sqrt{11} = 9\sqrt{11}$ m.
14.
Simplify $2\sqrt{63} + 3\sqrt{28} - \sqrt{112}$ and write the result as $\square\sqrt{7}$ . What number goes in the box?
Answer: 8
$2\sqrt{63} = 2 \cdot 3\sqrt{7} = 6\sqrt{7}$ , $3\sqrt{28} = 3 \cdot 2\sqrt{7} = 6\sqrt{7}$ , $\sqrt{112} = 4\sqrt{7}$ . So $6\sqrt{7} + 6\sqrt{7} - 4\sqrt{7} = 8\sqrt{7}$ .
15.
Simplify: $\sqrt{125} + \sqrt{80} - \sqrt{45} + \sqrt{20}$
Answer: $8\sqrt{5}$
1. Simplify: $\sqrt{125} = \square\sqrt{5}$ . What number goes in the box? 5
2. Simplify: $\sqrt{80} = \square\sqrt{5}$ . What number goes in the box? 4
3. Simplify: $\sqrt{45} = \square\sqrt{5}$ . What number goes in the box? 3
4. Simplify: $\sqrt{20} = \square\sqrt{5}$ . What number goes in the box? 2
5. Combine: $5\sqrt{5} + 4\sqrt{5} - 3\sqrt{5} + 2\sqrt{5} = \square\sqrt{5}$ . What number goes in the box? 8
16.
Simplify: $\sqrt{54} + \sqrt{24} - \sqrt{96}$
Answer: $\sqrt{6}$
1. Simplify: $\sqrt{54} = \square\sqrt{6}$ . What number goes in the box? 3
2. Simplify: $\sqrt{24} = \square\sqrt{6}$ . What number goes in the box? 2
3. Simplify: $\sqrt{96} = \square\sqrt{6}$ . What number goes in the box? 4
4. Combine: $3\sqrt{6} + 2\sqrt{6} - 4\sqrt{6} = \square\sqrt{6}$ . What number goes in the box? 1