Question

1. simplify the expression.

\sqrt{9e^{6x}} = \square

Explanation:

Step1: Use square - root property

We know that for any non - negative real numbers \(a\) and \(b\), \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\). So, for the expression \(\sqrt{9e^{6x}}\), we can split it as \(\sqrt{9}\cdot\sqrt{e^{6x}}\).

Step2: Simplify each square - root

We know that \(\sqrt{9} = 3\) because \(3\times3 = 9\). For \(\sqrt{e^{6x}}\), we use the property of exponents and square - roots: \(\sqrt{e^{y}}=e^{\frac{y}{2}}\). Here \(y = 6x\), so \(\sqrt{e^{6x}}=e^{\frac{6x}{2}}=e^{3x}\).

Step3: Combine the results

Multiply the two simplified square - roots together: \(3\times e^{3x}=3e^{3x}\).

Answer:

\(3e^{3x}\)