Problem

4 Show that $\frac{40-\sqrt{180}}{4 \sqrt{5}-3}$ can be written in the form $a \sqrt{b}$ where $a$ and $b$ are integers.

Answer

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Answer

\(\boxed{2\sqrt{5}}\) is the final answer

Steps

Step 1 ：Given expression: \(\frac{40-\sqrt{180}}{4\sqrt{5}-3}\)

Step 2 ：Rewrite \(\sqrt{180}\) as \(6\sqrt{5}\): \(\frac{40-6\sqrt{5}}{4\sqrt{5}-3}\)

Step 3 ：Multiply the numerator and denominator by the conjugate of the denominator: \(\frac{(40-6\sqrt{5})(4\sqrt{5}+3)}{(4\sqrt{5}-3)(4\sqrt{5}+3)}\)

Step 4 ：Simplify the expression: \(2\sqrt{5}\)

Step 5 ：\(\boxed{2\sqrt{5}}\) is the final answer

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