# How do you simplify sqrt3 - sqrt27 + 5sqrt12 ?

Mar 21, 2018

$8 \sqrt{3}$

#### Explanation:

$\sqrt{3} - \sqrt{27} + 5 \sqrt{12}$
$\sqrt{3} - \sqrt{9 \cdot 3} + 5 \sqrt{12}$ $\textcolor{b l u e}{\text{ 27 factors into } 9 \cdot 3}$
$\sqrt{3} - 3 \sqrt{3} + 5 \sqrt{12}$ $\textcolor{b l u e}{\text{ 9 is a perfect square, so take a 3 out}}$
$\sqrt{3} - 3 \sqrt{3} + 5 \sqrt{4 \cdot 3}$ $\textcolor{b l u e}{\text{ 12 factors into } 4 \cdot 3}$
$\sqrt{3} - 3 \sqrt{3} + 5 \cdot 2 \sqrt{3}$ $\textcolor{b l u e}{\text{ 4 is a perfect square, so take a 2 out}}$
$\sqrt{3} - 3 \sqrt{3} + 10 \sqrt{3}$ $\textcolor{b l u e}{\text{ To simplify, } 5 \cdot 2 = 10}$

Now that everything is in like terms of $\sqrt{3}$, we can simplify:

$\sqrt{3} - 3 \sqrt{3} + 10 \sqrt{3}$
$- 2 \sqrt{3} + 10 \sqrt{3}$ $\textcolor{b l u e}{\text{ Subtraction: } 1 \sqrt{3} - 3 \sqrt{3} = - 2 \sqrt{3}}$
$8 \sqrt{3}$ $\textcolor{b l u e}{\text{ Addition: } 10 \sqrt{3} + \left(- 2 \sqrt{3}\right) = 8 \sqrt{3}}$

Mar 21, 2018

√3−√27+5√12

=8√3

#### Explanation:

√3−√27+5√12
=√3−3√3+5√12
=√3−3√3+10√3
=8√3

• Simplify each surd to create a 'like' surd, when each number under the root sign is the same. This allows us to calculate the addition of the surds.
• We first simplify √27 to 9√3 = √27 and then simplify the number outside the root sign to = 3 (The square root) this gives us 3√3
• Then we simplify 5√12 to the √12 = 2√3 and then multiply this by 5 = 10√3
• Because each surd is now in the 'like' surd form we can carry out simple addition to complete the equation.
• =√3−3√3+10√3
=8√3
Mar 21, 2018

$8 \sqrt{3}$

#### Explanation:

Given: $\sqrt{3} - \sqrt{27} + 5 \sqrt{12}$

Simplify using perfect squares and the rule: $\sqrt{m \cdot n} = \sqrt{m} \cdot \sqrt{n}$

Some perfect squares are:
${2}^{2} = 4$
${3}^{2} = 9$
${4}^{2} = 16$
${5}^{2} = 25$
${6}^{2} = 36$
...

$\sqrt{3} - \sqrt{27} + 5 \sqrt{12}$

$= \sqrt{3} - \sqrt{9 \cdot 3} + 5 \sqrt{4 \cdot 3}$

$= \sqrt{3} - \sqrt{9} \sqrt{3} + 5 \sqrt{4} \sqrt{3}$

$= \sqrt{3} - 3 \sqrt{3} + 5 \cdot 2 \sqrt{3}$

$= \sqrt{3} - 3 \sqrt{3} + 10 \sqrt{3}$

Since all terms are alike they can be added or subtracted:

$\sqrt{3} - \sqrt{27} + 5 \sqrt{12} = 8 \sqrt{3}$