Question

complete the statement using <, >, or =
4√90 3√80

Explanation:

Step1: Simplify \(4\sqrt{90}\)

First, simplify the square root part. We know that \(\sqrt{90}=\sqrt{9\times10} = 3\sqrt{10}\) (since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a = 9,b = 10\) and \(\sqrt{9}=3\)). Then, \(4\sqrt{90}=4\times3\sqrt{10}=12\sqrt{10}\). Now, square \(12\sqrt{10}\) to make comparison easier (since both numbers are positive, squaring preserves the inequality). \((12\sqrt{10})^2=12^2\times(\sqrt{10})^2 = 144\times10=1440\).

Step2: Simplify \(3\sqrt{80}\)

Simplify \(\sqrt{80}=\sqrt{16\times5}=4\sqrt{5}\) (since \(\sqrt{16} = 4\)). Then, \(3\sqrt{80}=3\times4\sqrt{5}=12\sqrt{5}\). Square \(12\sqrt{5}\): \((12\sqrt{5})^2=12^2\times(\sqrt{5})^2=144\times5 = 720\). Wait, no, wait. Wait, actually, a better way is to square both original expressions directly. Wait, \( (4\sqrt{90})^2=16\times90=1440\), and \( (3\sqrt{80})^2=9\times80 = 720\). Wait, no, that's a mistake. Wait, no: \((a\sqrt{b})^2=a^2\times b\). So for \(4\sqrt{90}\), \(a = 4\), \(b = 90\), so \((4\sqrt{90})^2=16\times90 = 1440\). For \(3\sqrt{80}\), \(a = 3\), \(b = 80\), so \((3\sqrt{80})^2=9\times80=720\). Since \(1440>720\), and both \(4\sqrt{90}\) and \(3\sqrt{80}\) are positive, then \(4\sqrt{90}>3\sqrt{80}\).

Answer:

\(4\sqrt{90}>3\sqrt{80}\)