Question

simplify. assume x is greater than or equal to zero.\\(sqrt{75x^{9}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(75x^9\) into perfect square factors and other factors.

• For the coefficient: \(75 = 25\times3\), where \(25\) is a perfect square.
• For the variable part: \(x^9=x^{8 + 1}=x^{8}\times x\), where \(x^{8}=(x^{4})^{2}\) is a perfect square.

So, \(\sqrt{75x^{9}}=\sqrt{25\times3\times x^{8}\times x}\)

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\))

We can split the square root of the product into the product of square roots:
\(\sqrt{25\times3\times x^{8}\times x}=\sqrt{25}\times\sqrt{3}\times\sqrt{x^{8}}\times\sqrt{x}\)

Step3: Simplify each square root

• \(\sqrt{25} = 5\) (since \(5\times5 = 25\))
• \(\sqrt{x^{8}}=x^{4}\) (using the property \(\sqrt{x^{n}}=x^{\frac{n}{2}}\), here \(n = 8\), so \(\frac{8}{2}=4\))
• \(\sqrt{3}\) remains as it is, and \(\sqrt{x}\) remains as it is (since \(x\geq0\))

Putting it all together:
\(5\times x^{4}\times\sqrt{3}\times\sqrt{x}\)

Step4: Combine the square roots of the non - perfect square factors

Using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) again, \(\sqrt{3}\times\sqrt{x}=\sqrt{3x}\)

So the simplified form is \(5x^{4}\sqrt{3x}\)

Answer:

\(5x^{4}\sqrt{3x}\)