Question

simplify. assume v is greater than or equal to zero.
$3\sqrt{125v^{7}}$

Explanation:

Step1: Simplify the square root of 125

We know that \(125 = 25\times5\), and \(\sqrt{25}=5\), so \(\sqrt{125}=\sqrt{25\times5} = 5\sqrt{5}\).

Step2: Simplify the square root of \(v^{7}\)

Using the property of exponents \(\sqrt{a^{m}}=a^{\frac{m}{2}}\) (for \(a\geq0\)), for \(v^{7}\), we can write \(v^{7}=v^{6}\times v\), and \(\sqrt{v^{6}} = v^{3}\) (since \(v\geq0\)), so \(\sqrt{v^{7}}=\sqrt{v^{6}\times v}=v^{3}\sqrt{v}\).

Step3: Combine the results

Now, \(3\sqrt{125v^{7}}=3\times\sqrt{125}\times\sqrt{v^{7}}\). Substituting the values from Step 1 and Step 2, we get \(3\times5\sqrt{5}\times v^{3}\sqrt{v}\).
Multiplying the coefficients and using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) (for \(a,b\geq0\)), we have \(15v^{3}\sqrt{5v}\).

Answer:

\(15v^{3}\sqrt{5v}\)