# How do you simplify sqrt(20d) + 4sqrt(12d) - 3 sqrt(45d)?

Aug 15, 2017

$= 8 \sqrt{3 d} - 7 \sqrt{5 d}$

#### Explanation:

Write each radicand as the product of its factors and try to find squares wherever possible:

$\sqrt{20 d} + 4 \sqrt{12 d} - 3 \sqrt{45 d}$

$= \sqrt{4 \times 5 d} + 4 \sqrt{4 \times 3 d} - 3 \sqrt{9 \times 5 d} \text{ } \leftarrow$ find the roots

$= 2 \sqrt{5 d} + 4 \times 2 \sqrt{3 d} - 3 \times 3 \sqrt{5 d}$

$= 2 \sqrt{5 d} + 8 \sqrt{3 d} - 9 \sqrt{5 d} \text{ } \leftarrow$ collect like terms

$= 8 \sqrt{3 d} - 7 \sqrt{5 d}$

Aug 15, 2017

After simplifying, you are left with $8 \sqrt{3 d} - 7 \sqrt{5 d}$

#### Explanation:

First, let's simplify the numbers under the square root.

$\sqrt{20}$

This can be rewritten as $\sqrt{4 \cdot 5}$

4 has a square root, so we can pull its square root out of the radical, giving us $2 \sqrt{5}$

Now we do the same thing to $\sqrt{12}$ and $\sqrt{45}$

$\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}$

$\sqrt{45} = \sqrt{9 \cdot 5} = 3 \sqrt{5}$

We can't do much with the $\sqrt{d}$, so at this point we have:

$2 \sqrt{5 d} + 4 \cdot 2 \sqrt{3 d} - 3 \cdot 3 \sqrt{5 d}$

After multiplication, we have:

$2 \sqrt{5 d} + 8 \sqrt{3 d} - 9 \sqrt{5 d}$

We have two $\sqrt{5 d}$ terms, so we can combine them, giving us the final answer of:

$8 \sqrt{3 d} - 7 \sqrt{5 d}$