Simplify. Assume a and b are greater than zero.

√5ab^3/3

1 answer

To simplify the expression \(\sqrt{\frac{5ab^3}{3}}\), we can break it down as follows:

  1. Split the square root into two parts: \[ \sqrt{\frac{5ab^3}{3}} = \frac{\sqrt{5ab^3}}{\sqrt{3}} \]

  2. Now, simplify the square root of the numerator: \[ \sqrt{5ab^3} = \sqrt{5} \cdot \sqrt{a} \cdot \sqrt{b^3} \]

  3. Since \(b^3 = b^2 \cdot b\), we can simplify \(\sqrt{b^3}\): \[ \sqrt{b^3} = \sqrt{b^2 \cdot b} = \sqrt{b^2} \cdot \sqrt{b} = b \cdot \sqrt{b} \]

  4. Combine everything together: \[ \sqrt{5ab^3} = \sqrt{5} \cdot \sqrt{a} \cdot b \cdot \sqrt{b} = b\sqrt{5a} \cdot \sqrt{b} \] Therefore: \[ \sqrt{5ab^3} = b\sqrt{5ab} \]

  5. Now, put it all back into the expression: \[ \frac{\sqrt{5ab^3}}{\sqrt{3}} = \frac{b\sqrt{5ab}}{\sqrt{3}} \]

The final simplified expression is: \[ \frac{b\sqrt{5ab}}{\sqrt{3}} \]