Question

Is the radical expression in simplest form? Explain. a. \(\frac{3}{5} \sqrt{2}\) b. \(\sqrt{\frac{3}{16}}\) c. \(5 \sqrt{40}\)

Short answer

a. Yes, the expression is already in its simplest form. b. Yes, the simplified form is \(\frac{\sqrt{3}}{4}\). c. No, it can be simplified to \(10\sqrt{10}\).

Step-by-step solution

- Simplification of Expression a

Look at the first expression \(\frac{3}{5} \sqrt{2}\). It's already in simplest form because the number under the square root, \(2\), cannot be simplified further. Furthermore, the coefficient of the square root is a fraction which is already simplified.

- Simplification of Expression b

The second expression is \(\sqrt{\frac{3}{16}}\). To simplify a fraction under a square root, take the square root of the numerator and the denominator separately if possible. So, \(\sqrt{\frac{3}{16}} = \frac{\sqrt{3}}{\sqrt{16}} = \frac{\sqrt{3}}{4}\). The fraction \(\frac{\sqrt{3}}{4}\) is already in simplest form since \(\sqrt{3}\) can't be simplified further.

- Simplification of Expression c

The third expression is \(5 \sqrt{40}\). The number under the square root, \(40\), can be simplified by finding a pair of identical factors. 40 can be factored into \(4*10\), and the square root of \(4\) is \(2\). So, we can simplify \(5 \sqrt{40}\) to \(5*2\sqrt{10}\) which equals \(10\sqrt{10}\).