Question

simplify. assume a is greater than or equal to zero.
$2\sqrt{18a^8}$

Explanation:

Step1: Factor the radicand

Factor \(18a^8\) into perfect squares and other factors. We know that \(18 = 9\times2\) and \(a^8=(a^4)^2\). So, \(18a^8=9\times2\times(a^4)^2\). Then the expression becomes \(2\sqrt{9\times2\times(a^4)^2}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)) and \(\sqrt{x^2}=x\) (for \(x\geq0\)), we can split the square root: \(\sqrt{9\times2\times(a^4)^2}=\sqrt{9}\times\sqrt{2}\times\sqrt{(a^4)^2}\). Since \(\sqrt{9} = 3\) and \(\sqrt{(a^4)^2}=a^4\) (because \(a\geq0\)), this simplifies to \(3\times\sqrt{2}\times a^4\).

Step3: Multiply with the coefficient outside

Now, multiply this with the coefficient \(2\) outside the square root: \(2\times3\times\sqrt{2}\times a^4\).

Step4: Simplify the product

\(2\times3 = 6\), so the simplified expression is \(6a^4\sqrt{2}\).

Answer:

\(6a^{4}\sqrt{2}\)