Question

9 what is the simplest form of the expression? \\( \sqrt3{108a^{16}b^9} \\) a \\( 3a^5b^3 \sqrt3{4a} \\) b none of these c \\( 4a^5b^3 \sqrt3{3a} \\) d \\( 3a^5b \sqrt3{a} \\)

Explanation:

Step1: Factor the radicand

Factor \(108\), \(a^{16}\), and \(b^9\) into parts that are perfect cubes (since it's a cube root).

• \(108 = 27\times4 = 3^3\times4\)
• \(a^{16}=a^{15 + 1}=a^{3\times5}\times a=(a^5)^3\times a\)
• \(b^9=(b^3)^3\)

So, \(\sqrt[3]{108a^{16}b^9}=\sqrt[3]{3^3\times4\times(a^5)^3\times a\times(b^3)^3}\)

Step2: Apply the cube root property

The cube root property is \(\sqrt[3]{xyz}=\sqrt[3]{x}\cdot\sqrt[3]{y}\cdot\sqrt[3]{z}\) (for real numbers \(x,y,z\)). Also, \(\sqrt[3]{x^3}=x\).

Using these properties:

$$
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$$

Answer:

A. \(3a^5b^3\sqrt[3]{4a}\)