Question

simplify $4\sqrt{3} + 5\sqrt{12}$. the simplified expression is \boxed{}

Explanation:

Step1: Simplify $\sqrt{12}$

We know that $12 = 4\times3$, so $\sqrt{12}=\sqrt{4\times3}$. By the property of square roots $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (where $a = 4$, $b = 3$ and $a\geq0,b\geq0$), we have $\sqrt{4\times3}=\sqrt{4}\times\sqrt{3}=2\sqrt{3}$.

Step2: Substitute and simplify the expression

Substitute $\sqrt{12}=2\sqrt{3}$ into the original expression $4\sqrt{3}+5\sqrt{12}$, we get $4\sqrt{3}+5\times(2\sqrt{3})$.
First, calculate $5\times(2\sqrt{3}) = 10\sqrt{3}$.
Then, the expression becomes $4\sqrt{3}+10\sqrt{3}$.
Since $4\sqrt{3}$ and $10\sqrt{3}$ are like terms (they have the same radical part $\sqrt{3}$), we can combine them by adding the coefficients: $(4 + 10)\sqrt{3}=14\sqrt{3}$.

Answer:

$14\sqrt{3}$