How do you simplify 2√50 + 12√8?

First, find perfect square factors: 50=25*2 and 8=4*2. Simplify the radicals: 2(5√2) + 12(2√2) = 10√2 + 24√2. Finally, combine the like terms to get 34√2.

Why can't you simplify √x + √y as √x+y?

The square root is not distributive over addition or subtraction. For example, √16 + √9 = 4 + 3 = 7, but √(16+9) = √25 = 5. Since 7 does not equal 5, the rule does not apply.

Simplify Radicals Practice Problems

Simplifying radicals is a foundational algebra skill. Remember that to simplify a radical, you must find the largest perfect square factor of the radicand (the number under the radical). Detailed, pedagogical solutions are provided below to help you understand the mechanics of each step.

Practice Questions

Question 1

Simplify: \( 2 \sqrt{50} + 12 \sqrt{8} \)

A) \(14 \sqrt{58}\) B) 240 C) 280 D) 98 E) \(34 \sqrt{2}\)

Show Detailed Solution

1. Find perfect squares: \(50 = 25 \cdot 2\) and \(8 = 4 \cdot 2\).

2. Extract roots: \(2\sqrt{25 \cdot 2} + 12\sqrt{4 \cdot 2} = 2(5)\sqrt{2} + 12(2)\sqrt{2}\).

3. Simplify: \(10\sqrt{2} + 24\sqrt{2}\).

4. Since the radicands (\(\sqrt{2}\)) match, add the coefficients: \(10 + 24 = 34\). Result: \(\mathbf{34\sqrt{2}}\).

Question 2

Simplify: \( \sqrt{27} - \sqrt{300} \)

A) \(\sqrt{-273}\) B) \(-7 \sqrt{3}\) C) -147 D) \(3 - 10 \sqrt{3}\) E) \(7\sqrt{3}\)

Show Detailed Solution

1. Identify perfect square factors: \(27 = 9 \cdot 3\) and \(300 = 100 \cdot 3\).

2. Simplify each term: \(\sqrt{9}\sqrt{3} - \sqrt{100}\sqrt{3} = 3\sqrt{3} - 10\sqrt{3}\).

3. Combine like terms: \((3 - 10)\sqrt{3} = \mathbf{-7\sqrt{3}}\).

Question 3

Simplify: \( - 2 \sqrt{16y} + 10 \sqrt{y} \)

A) \(8 \sqrt{y}\) B) \(8 \sqrt{-15y}\) C) \(8 \sqrt{17y}\) D) \(2 \sqrt{16y}\) E) \(2 \sqrt{y}\)

Show Detailed Solution

1. Simplify the radical: \(\sqrt{16y} = \sqrt{16} \cdot \sqrt{y} = 4\sqrt{y}\).

2. Substitute back: \(-2(4\sqrt{y}) + 10\sqrt{y} = -8\sqrt{y} + 10\sqrt{y}\).

3. Combine: \((-8 + 10)\sqrt{y} = \mathbf{2\sqrt{y}}\).

Question 4

Simplify: \( 2 \sqrt{x + 1} + 3 \sqrt{16x + 16} \)

A) \(14 \sqrt{x + 1}\) B) \(5 \sqrt{17x + 17}\) C) 5 D) \(\sqrt{17x + 17}\) E) \(6 \sqrt{17x + 17}\)

Show Detailed Solution

1. Factor the second radicand: \(16x + 16 = 16(x+1)\).

2. Apply root rule: \(\sqrt{16(x+1)} = \sqrt{16}\sqrt{x+1} = 4\sqrt{x+1}\).

3. Multiply: \(3 \cdot 4\sqrt{x+1} = 12\sqrt{x+1}\).

4. Add: \(2\sqrt{x+1} + 12\sqrt{x+1} = \mathbf{14\sqrt{x+1}}\).

Question 5

\( 2 \sqrt{3} + 4 \sqrt{12} + 3 \sqrt{48} = \)

A) \(9 \sqrt{3}\) B) \(9 \sqrt{63}\) C) 22 D) \(22 \sqrt{3}\) E) \(9 \sqrt{48}\)

Show Detailed Solution

1. Factor radicals: \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\) and \(\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}\).

2. Multiply coefficients: \(2\sqrt{3} + 4(2\sqrt{3}) + 3(4\sqrt{3}) = 2\sqrt{3} + 8\sqrt{3} + 12\sqrt{3}\).

3. Sum coefficients: \(2 + 8 + 12 = 22\). Result: \(\mathbf{22\sqrt{3}}\).

Question 6

Rewrite: \( \dfrac{\sqrt{3} + \sqrt{12}}{\sqrt{3} - \sqrt{12}} \) without radicals.

A) 1 B) 0 C) -3 D) -1 E) 3

Show Detailed Solution

1. Simplify \(\sqrt{12}\): \(\sqrt{4 \cdot 3} = 2\sqrt{3}\).

2. Rewrite expression: \(\frac{\sqrt{3} + 2\sqrt{3}}{\sqrt{3} - 2\sqrt{3}}\).

3. Combine like terms: \(\frac{3\sqrt{3}}{-1\sqrt{3}}\).

4. Cancel \(\sqrt{3}\): \(3 / -1 = \mathbf{-3}\).

Question 7

Simplify: \( 5 \sqrt{x} + 6 \sqrt{9x} - 10 \sqrt{16x} \)

A) \(\sqrt{x}\) B) \(-17 \sqrt{x}\) C) \(\sqrt{46x}\) D) \(-2 \sqrt{x}\) E) \(\sqrt{-25x}\)

Show Detailed Solution

1. Simplify radicals: \(6\sqrt{9x} = 6 \cdot 3\sqrt{x} = 18\sqrt{x}\) and \(10\sqrt{16x} = 10 \cdot 4\sqrt{x} = 40\sqrt{x}\).

2. Combine: \(5\sqrt{x} + 18\sqrt{x} - 40\sqrt{x}\).

3. Calculate: \((5 + 18 - 40)\sqrt{x} = \mathbf{-17\sqrt{x}}\).

Question 8

\( 2 \sqrt{27} + 2 \sqrt{75} = \)

A) \(16 \sqrt{3}\) B) \(4 \sqrt{3}\) C) \(4 \sqrt{102}\) D) 16 E) 204

Show Detailed Solution

1. Simplify: \(\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}\) and \(\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}\).

2. Apply coefficients: \(2(3\sqrt{3}) + 2(5\sqrt{3}) = 6\sqrt{3} + 10\sqrt{3}\).

3. Combine: \(\mathbf{16\sqrt{3}}\).

Question 9

\( \sqrt{10^3} + \sqrt{10^5} = \)

A) \(\sqrt{10,100}\) B) \(110 \sqrt{10}\) C) 10,000 D) \(2 \sqrt{1,000}\) E) \(2 \sqrt{100,000}\)

Show Detailed Solution

1. Break down powers of 10 into a square and a remainder: \(10^3 = 10^2 \cdot 10\) and \(10^5 = 10^4 \cdot 10\).

2. Root extraction: \(\sqrt{10^2}\sqrt{10} + \sqrt{10^4}\sqrt{10} = 10\sqrt{10} + 100\sqrt{10}\).

3. Sum: \(\mathbf{110\sqrt{10}}\).

Question 10

Simplify \( \sqrt{8} \sqrt{3} \sqrt{6} \) without radicals.

A) 144 B) 3 C) 17 D) 12 E) 4

Show Detailed Solution

1. Combine under one radical using the product rule: \(\sqrt{8 \cdot 3 \cdot 6}\).

2. Multiply the numbers: \(8 \cdot 3 = 24\) and \(24 \cdot 6 = 144\).

3. Solve: \(\sqrt{144} = \mathbf{12}\).