Question

simplify.
\sqrt{2x} \cdot \sqrt{11x}
assume that the variable represents a positive real number.

Explanation:

Step1: Use product rule of square roots

The product rule for square roots states that \(\sqrt{a} \cdot \sqrt{b}=\sqrt{ab}\) for non - negative real numbers \(a\) and \(b\). Here, \(a = 2x\) and \(b=11x\), so \(\sqrt{2x}\cdot\sqrt{11x}=\sqrt{(2x)\cdot(11x)}\).

Step2: Multiply the terms inside the square root

Multiply the coefficients and the variables separately. The coefficient of \(x\) in \(2x\) is \(2\) and in \(11x\) is \(11\), so \(2\times11 = 22\). For the variable \(x\), using the rule of exponents \(x^m\cdot x^n=x^{m + n}\), here \(m = 1\) and \(n = 1\), so \(x\cdot x=x^{1 + 1}=x^{2}\). Thus, \((2x)\cdot(11x)=22x^{2}\), and our expression becomes \(\sqrt{22x^{2}}\).

Step3: Simplify the square root

We know that \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a\geq0,b\geq0\)). So \(\sqrt{22x^{2}}=\sqrt{22}\cdot\sqrt{x^{2}}\). Since \(x\) is a positive real number, \(\sqrt{x^{2}}=x\). So \(\sqrt{22}\cdot\sqrt{x^{2}}=x\sqrt{22}\).

Answer:

\(x\sqrt{22}\)