Question

g. $5sqrt{20} - 3sqrt{5} - 4sqrt{24}$

Explanation:

Step1: Simplify each radical

Simplify \( \sqrt{20} \) and \( \sqrt{24} \). We know that \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) for \( a\geq0, b\geq0 \).
For \( \sqrt{20} \), we can write \( 20 = 4\times5 \), so \( \sqrt{20}=\sqrt{4\times5}=\sqrt{4}\cdot\sqrt{5}=2\sqrt{5} \).
For \( \sqrt{24} \), we can write \( 24 = 4\times6 \), so \( \sqrt{24}=\sqrt{4\times6}=\sqrt{4}\cdot\sqrt{6}=2\sqrt{6} \).

Step2: Substitute the simplified radicals back

Substitute \( \sqrt{20}=2\sqrt{5} \) and \( \sqrt{24}=2\sqrt{6} \) into the original expression \( 5\sqrt{20}-3\sqrt{5}-4\sqrt{24} \).
We get \( 5\times(2\sqrt{5})-3\sqrt{5}-4\times(2\sqrt{6}) \).

Step3: Perform the multiplication

Calculate the multiplications: \( 5\times(2\sqrt{5}) = 10\sqrt{5} \) and \( 4\times(2\sqrt{6}) = 8\sqrt{6} \).
So the expression becomes \( 10\sqrt{5}-3\sqrt{5}-8\sqrt{6} \).

Step4: Combine like terms

Combine the terms with \( \sqrt{5} \). \( 10\sqrt{5}-3\sqrt{5}=(10 - 3)\sqrt{5}=7\sqrt{5} \).
The term with \( \sqrt{6} \) remains as it is since there are no other like terms.

Answer:

\( 7\sqrt{5}-8\sqrt{6} \)