(sqrt (450x^4))/(sqrt (10x))= square

To simplify the expression \(\frac{\sqrt{450 x^{4}}}{\sqrt{10 x}}\), we can start by simplifying each square root separately.<br /><br />First, consider \(\sqrt{450 x^4}\):<br />- Break down 450 into its prime factors: \(450 = 2 \times 3^2 \times 5^2\).<br />- Therefore, \(\sqrt{450} = \sqrt{2 \times 3^2 \times 5^2} = \sqrt{2} \times 3 \times 5 = 15\sqrt{2}\).<br /><br />Now, for the variable part:<br />- \(\sqrt{x^4} = x^2\) because the square root of \(x^4\) is \(x^2\).<br /><br />So, \(\sqrt{450 x^4} = 15\sqrt{2} \cdot x^2 = 15x^2\sqrt{2}\).<br /><br />Next, consider \(\sqrt{10 x}\):<br />- Break down 10 into its prime factors: \(10 = 2 \times 5\).<br />- Therefore, \(\sqrt{10} = \sqrt{2 \times 5} = \sqrt{10}\).<br /><br />For the variable part:<br />- \(\sqrt{x} = x^{1/2}\).<br /><br />So, \(\sqrt{10 x} = \sqrt{10} \cdot x^{1/2}\).<br /><br />Now, substitute these simplified forms back into the original expression:<br /><br />\[<br />\frac{\sqrt{450 x^4}}{\sqrt{10 x}} = \frac{15x^2\sqrt{2}}{\sqrt{10} \cdot x^{1/2}}<br />\]<br /><br />Simplify the expression:<br /><br />1. Simplify the coefficients: <br /> - \(\frac{15\sqrt{2}}{\sqrt{10}} = \frac{15\sqrt{2}}{\sqrt{2 \times 5}} = \frac{15\sqrt{2}}{\sqrt{2}\sqrt{5}} = \frac{15}{\sqrt{5}}\).<br /><br />2. Rationalize the denominator:<br /> - Multiply numerator and denominator by \(\sqrt{5}\): <br /> \(\frac{15}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{15\sqrt{5}}{5} = 3\sqrt{5}\).<br /><br />3. Simplify the variable part:<br /> - \(\frac{x^2}{x^{1/2}} = x^{2 - 1/2} = x^{3/2}\).<br /><br />Combine these results:<br /><br />\[<br />\frac{\sqrt{450 x^4}}{\sqrt{10 x}} = 3\sqrt{5} \cdot x^{3/2}<br />\]<br /><br />Thus, the simplified form of the expression is:<br /><br />\[<br />3x^{3/2}\sqrt{5}<br />\]