Question

simplify. \\(\sqrt{27} + 4\sqrt{75}\\)

Explanation:

Step1: Simplify $\sqrt{27}$

We know that $27 = 9\times3$, and $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (where $a\geq0,b\geq0$). So $\sqrt{27}=\sqrt{9\times3}=\sqrt{9}\times\sqrt{3}=3\sqrt{3}$.

Step2: Simplify $4\sqrt{75}$

First, factor $75$ as $25\times3$. Then, using the property $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (where $a\geq0,b\geq0$), we have $\sqrt{75}=\sqrt{25\times3}=\sqrt{25}\times\sqrt{3}=5\sqrt{3}$. So $4\sqrt{75}=4\times5\sqrt{3}=20\sqrt{3}$.

Step3: Add the simplified radicals

Now we add $3\sqrt{3}$ and $20\sqrt{3}$. Since they are like radicals (same radicand), we add the coefficients: $3\sqrt{3}+20\sqrt{3}=(3 + 20)\sqrt{3}=23\sqrt{3}$.

Answer:

$23\sqrt{3}$