Question

\\6ab \sqrt{\frac{32a^5}{120b^3}}\\

Explanation:

Simplify the fraction inside the radical

First, we simplify the fraction under the square root by dividing the coefficients and reducing variables where possible.

$$ \frac{32a^5}{120b^3} = \frac{4a^5}{15b^3} $$

Rewrite the expression with separated radicals

Next, we apply the quotient property of radicals to separate the numerator and the denominator.

$$ 6ab \sqrt{\frac{4a^5}{15b^3}} = 6ab \frac{\sqrt{4a^5}}{\sqrt{15b^3}} $$

Simplify the square roots in the numerator and denominator

We simplify the perfect squares out of the radicals. We assume variables represent positive real numbers.

$$ \sqrt{4a^5} = \sqrt{4 \cdot a^4 \cdot a} = 2a^2\sqrt{a} $$

$$ \sqrt{15b^3} = \sqrt{b^2 \cdot 15b} = b\sqrt{15b} $$

Substitute these back into our expression:

$$ 6ab \cdot \frac{2a^2\sqrt{a}}{b\sqrt{15b}} = \frac{12a^3b\sqrt{a}}{b\sqrt{15b}} = \frac{12a^3\sqrt{a}}{\sqrt{15b}} $$

Rationalize the denominator

To eliminate the radical in the denominator, we multiply the numerator and the denominator by \(\sqrt{15b}\).

$$ \frac{12a^3\sqrt{a} \cdot \sqrt{15b}}{\sqrt{15b} \cdot \sqrt{15b}} = \frac{12a^3\sqrt{15ab}}{15b} $$

Reduce the final fraction

We simplify the fraction by dividing the numerator and denominator coefficients by their greatest common divisor, which is 3.

$$ \frac{12a^3\sqrt{15ab}}{15b} = \frac{4a^3\sqrt{15ab}}{5b} $$

Answer:

$$\frac{4a^3\sqrt{15ab}}{5b}$$