Question

which expression is equivalent to (sqrt3{\frac{1}{1000} c^9 d^{12}}) ?
options:
(\frac{1}{100} c^3 d^4)
(\frac{1}{100} c^6 d^9)
(\frac{1}{10} c^3 d^4)
(\frac{1}{10} c^6 d^9)

Explanation:

Step1: Recall Cube Root Properties

For a cube root \(\sqrt[3]{x^n}\), we can use the property \(\sqrt[3]{x^n}=x^{\frac{n}{3}}\). Also, for a fraction \(\sqrt[3]{\frac{a}{b}}=\frac{\sqrt[3]{a}}{\sqrt[3]{b}}\).

Step2: Simplify the Coefficient

First, simplify \(\sqrt[3]{\frac{1}{1000}}\). Since \(1000 = 10^3\), \(\sqrt[3]{\frac{1}{10^3}}=\frac{1}{10}\) (because \(\sqrt[3]{x^3}=x\) for real \(x\)).

Step3: Simplify the Variables

Now, for the variables \(c^9d^{12}\), apply the cube - root property. For \(c^9\): \(\sqrt[3]{c^9}=c^{\frac{9}{3}} = c^3\)? Wait, no, wait. Wait, the original expression is \(\sqrt[3]{\frac{1}{1000}c^9d^{12}}\). Let's re - do that.

Wait, \(\sqrt[3]{\frac{1}{1000}c^{9}d^{12}}=\sqrt[3]{\frac{1}{10^3}\times c^{9}\times d^{12}}\). Using the property \(\sqrt[3]{a\times b\times c}=\sqrt[3]{a}\times\sqrt[3]{b}\times\sqrt[3]{c}\), we have:

\(\sqrt[3]{\frac{1}{10^3}}\times\sqrt[3]{c^{9}}\times\sqrt[3]{d^{12}}\)

We know that \(\sqrt[3]{\frac{1}{10^3}}=\frac{1}{10}\) (because \((\frac{1}{10})^3=\frac{1}{1000}\)).

For \(\sqrt[3]{c^{9}}\), using the rule \(\sqrt[n]{x^m}=x^{\frac{m}{n}}\), here \(n = 3\) and \(m = 9\), so \(c^{\frac{9}{3}}=c^{3}\)? Wait, no, wait the options have \(c^{3}\) or \(c^{6}\)? Wait, no, looking at the options:

Wait the options are:

1. \(\frac{1}{100}c^{3}d^{4}\)

1. \(\frac{1}{100}c^{6}d^{9}\)

1. \(\frac{1}{10}c^{3}d^{4}\)

1. \(\frac{1}{10}c^{6}d^{9}\)

Wait, I made a mistake in the exponent calculation. Let's recalculate:

For \(\sqrt[3]{c^{9}}\): \(\frac{9}{3}=3\)? No, wait \(9\div3 = 3\), so \(c^{3}\)? But wait, no, wait the original exponent of \(c\) is \(9\), cube root of \(c^{9}\) is \(c^{9/3}=c^{3}\)? Wait no, \(3\times3 = 9\), so \((c^{3})^3=c^{9}\), so \(\sqrt[3]{c^{9}}=c^{3}\). For \(d^{12}\), \(\sqrt[3]{d^{12}}=d^{12/3}=d^{4}\) (because \((d^{4})^3=d^{12}\)). And \(\sqrt[3]{\frac{1}{1000}}=\frac{1}{10}\) (because \(10^3 = 1000\)).

So putting it all together: \(\sqrt[3]{\frac{1}{1000}c^{9}d^{12}}=\frac{1}{10}c^{3}d^{4}\)

Answer:

\(\frac{1}{10}c^{3}d^{4}\) (the third option)