Question

1. simplify the expression. (sqrt3{8c^{12x}}) =

Explanation:

Step1: Recall the property of cube roots

For a cube root \(\sqrt[3]{x^3}=x\), and \(\sqrt[3]{ab}=\sqrt[3]{a}\cdot\sqrt[3]{b}\) (where \(a = 8\), \(b = c^{12x}\) in this case). Also, \(8 = 2^3\).
So we can rewrite the expression \(\sqrt[3]{8c^{12x}}\) as \(\sqrt[3]{2^3\cdot c^{12x}}\).

Step2: Apply the cube - root property

Using the property \(\sqrt[3]{ab}=\sqrt[3]{a}\cdot\sqrt[3]{b}\), we have \(\sqrt[3]{2^3\cdot c^{12x}}=\sqrt[3]{2^3}\cdot\sqrt[3]{c^{12x}}\).
Since \(\sqrt[3]{2^3} = 2\) and for \(\sqrt[3]{c^{12x}}\), using the property \(\sqrt[n]{x^m}=x^{\frac{m}{n}}\) (here \(n = 3\) and \(m=12x\)), we get \(\sqrt[3]{c^{12x}}=c^{\frac{12x}{3}}=c^{4x}\).

Step3: Combine the results

Multiplying the two results together, we have \(2\cdot c^{4x}=2c^{4x}\).

Answer:

\(2c^{4x}\)