Question

express in simplest radical form given ( x > 0 ).( xsqrt{75x^{3}} + x^{2}sqrt{48x} )

Explanation:

Step1: Simplify each radical term

For \(x\sqrt{75x^{3}}\), first factor \(75x^{3}\): \(75x^{3}=25x^{2}\cdot3x\). Then \(\sqrt{75x^{3}}=\sqrt{25x^{2}\cdot3x}=\sqrt{25x^{2}}\cdot\sqrt{3x}=5x\sqrt{3x}\) (since \(x > 0\)). So \(x\sqrt{75x^{3}}=x\cdot5x\sqrt{3x}=5x^{2}\sqrt{3x}\).

For \(x^{2}\sqrt{48x}\), factor \(48x\): \(48x = 16\cdot3x\). Then \(\sqrt{48x}=\sqrt{16\cdot3x}=\sqrt{16}\cdot\sqrt{3x}=4\sqrt{3x}\). So \(x^{2}\sqrt{48x}=x^{2}\cdot4\sqrt{3x}=4x^{2}\sqrt{3x}\).

Step2: Combine like terms

Now we have \(5x^{2}\sqrt{3x}+4x^{2}\sqrt{3x}\). Since the radical parts are the same, we can add the coefficients: \((5 + 4)x^{2}\sqrt{3x}=9x^{2}\sqrt{3x}\).

Answer:

\(9x^{2}\sqrt{3x}\)