Question

1. $3sqrt{24x} - 2sqrt{54x} + sqrt{48}$
2. $5sqrt{18c} + 3sqrt{72c} + 6sqrt{76}$

Explanation:

Step1: Simplify each radical term

For \(3\sqrt{24x}\):
Factor \(24x = 4\times6x\), so \(\sqrt{24x}=\sqrt{4\times6x}=2\sqrt{6x}\). Then \(3\sqrt{24x}=3\times2\sqrt{6x}=6\sqrt{6x}\).
For \(-2\sqrt{54x}\):
Factor \(54x = 9\times6x\), so \(\sqrt{54x}=\sqrt{9\times6x}=3\sqrt{6x}\). Then \(-2\sqrt{54x}=-2\times3\sqrt{6x}=-6\sqrt{6x}\).
For \(\sqrt{48}\):
Factor \(48 = 16\times3\), so \(\sqrt{48}=\sqrt{16\times3}=4\sqrt{3}\).

Step2: Combine like terms

\(3\sqrt{24x}-2\sqrt{54x}+\sqrt{48}=6\sqrt{6x}-6\sqrt{6x}+4\sqrt{3}\)
The \(6\sqrt{6x}\) and \(-6\sqrt{6x}\) cancel out, leaving \(4\sqrt{3}\).

For problem 44:

Step1: Simplify each radical term

For \(5\sqrt{18c}\):
Factor \(18c = 9\times2c\), so \(\sqrt{18c}=\sqrt{9\times2c}=3\sqrt{2c}\). Then \(5\sqrt{18c}=5\times3\sqrt{2c}=15\sqrt{2c}\).
For \(3\sqrt{72c}\):
Factor \(72c = 36\times2c\), so \(\sqrt{72c}=\sqrt{36\times2c}=6\sqrt{2c}\). Then \(3\sqrt{72c}=3\times6\sqrt{2c}=18\sqrt{2c}\).
For \(6\sqrt{76}\):
Factor \(76 = 4\times19\), so \(\sqrt{76}=\sqrt{4\times19}=2\sqrt{19}\). Then \(6\sqrt{76}=6\times2\sqrt{19}=12\sqrt{19}\).

Step2: Combine like terms

\(5\sqrt{18c}+3\sqrt{72c}+6\sqrt{76}=15\sqrt{2c}+18\sqrt{2c}+12\sqrt{19}\)
Combine the like terms with \(\sqrt{2c}\): \(15\sqrt{2c}+18\sqrt{2c}=33\sqrt{2c}\). So the result is \(33\sqrt{2c}+12\sqrt{19}\).

Answer:

Problem 13: \(4\sqrt{3}\)
Problem 44: \(33\sqrt{2c}+12\sqrt{19}\)