Question

simplify. assume d is greater than or equal to zero. \\(sqrt{45d^{10}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(45d^{10}\) into a product of a perfect square and another number. We know that \(45 = 9\times5\) and \(d^{10}=(d^{5})^{2}\), so \(\sqrt{45d^{10}}=\sqrt{9\times5\times(d^{5})^{2}}\).

Step2: Use the property of square roots

The property of square roots states that \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a\geq0\) and \(b\geq0\)) and \(\sqrt{a^{2}} = a\) (where \(a\geq0\)). Applying these properties, we have \(\sqrt{9\times5\times(d^{5})^{2}}=\sqrt{9}\times\sqrt{(d^{5})^{2}}\times\sqrt{5}\). Since \(\sqrt{9} = 3\) and \(\sqrt{(d^{5})^{2}}=d^{5}\) (because \(d\geq0\)), this simplifies to \(3\times d^{5}\times\sqrt{5}\).

Step3: Simplify the expression

Multiplying the terms together, we get \(3d^{5}\sqrt{5}\).

Answer:

\(3d^{5}\sqrt{5}\)