Question

simplify.
\\(\sqrt{5c} \cdot \sqrt{7c}\\)
assume that the variable represents a positive real number.

Explanation:

Step1: Recall the property of square roots

The product of two square roots \(\sqrt{a} \cdot \sqrt{b}=\sqrt{ab}\) (for \(a\geq0,b\geq0\)). Here, \(a = 5c\) and \(b=7c\), and since \(c\) is a positive real number, \(5c\geq0\) and \(7c\geq0\). So we can apply this property: \(\sqrt{5c}\cdot\sqrt{7c}=\sqrt{(5c)\cdot(7c)}\).

Step2: Simplify the product inside the square root

Multiply the coefficients and the variables separately: \((5c)\cdot(7c)=5\times7\times c\times c = 35c^{2}\). So now we have \(\sqrt{35c^{2}}\).

Step3: Simplify the square root of \(c^{2}\)

Since \(c\) is a positive real number, \(\sqrt{c^{2}} = c\) (because if \(c>0\), then \(c^{2}\) square - rooted gives \(c\)). And \(\sqrt{35}\) cannot be simplified further. So \(\sqrt{35c^{2}}=\sqrt{35}\cdot\sqrt{c^{2}}=\sqrt{35}c = c\sqrt{35}\).

Answer:

\(c\sqrt{35}\)