Question

simplify. assume ( d ) is greater than or equal to zero. ( 5sqrt{12d^{3}} )

Explanation:

Step1: Factor the radicand

We can factor \(12d^3\) as \(4\times3\times d^2\times d\), since \(4\) is a perfect square and \(d^2\) is also a perfect square (given \(d\geq0\)). So, \(12d^3 = 4d^2\times3d\).

$$ 5\sqrt{12d^3}=5\sqrt{4d^2\times3d} $$

Step2: Use the property of square roots

The property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)) allows us to separate the perfect square factors from the non - perfect square factors. We know that \(\sqrt{4d^2}=\sqrt{4}\times\sqrt{d^2}\), and since \(d\geq0\), \(\sqrt{d^2} = d\) and \(\sqrt{4}=2\). So:

$$ 5\sqrt{4d^2\times3d}=5\times\sqrt{4d^2}\times\sqrt{3d}=5\times2d\times\sqrt{3d} $$

Step3: Simplify the coefficient

Multiply the coefficients \(5\) and \(2\):

$$ 5\times2d\times\sqrt{3d}=10d\sqrt{3d} $$

Answer:

\(10d\sqrt{3d}\)