# How do you multiply sqrt(2)(sqrt 8 + sqrt 4)?

Jun 29, 2015

$\sqrt{2} \left(\sqrt{8} + \sqrt{4}\right) = 4 + 2 \sqrt{2}$

#### Explanation:

Method 1: Multiply first then simplify
The distributive property (of multiplication over addition) tells us that
$\textcolor{w h i t e}{\text{XXXX}}$$a \cdot \left(b + c\right) = a b + a c$

Further as a property of exponents (and therefore of square roots)
$\textcolor{w h i t e}{\text{XXXX}}$${a}^{m} \cdot {b}^{m} = {\left(a \cdot b\right)}^{m}$
$\textcolor{w h i t e}{\text{XXXX}}$$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ since $\left(\sqrt{k} = {k}^{\frac{1}{2}}\right)$

So
$\sqrt{2} \left(\sqrt{8} + \sqrt{4}\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$= \sqrt{2} \cdot \sqrt{8} + \sqrt{2} \cdot \sqrt{4}$

$\textcolor{w h i t e}{\text{XXXX}}$$= \sqrt{2 \cdot 8} + \sqrt{2 \cdot 4}$

$\textcolor{w h i t e}{\text{XXXX}}$$= \sqrt{16} + \sqrt{8}$

$\textcolor{w h i t e}{\text{XXXX}}$$= 4 + 2 \sqrt{2}$

Method 2: Simplify roots then multiply
$\sqrt{2} \left(\sqrt{8} + \sqrt{4}\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$= \sqrt{2} \left(2 \sqrt{2} + 2\right)$

$\textcolor{w h i t e}{\text{XXXX}}$$= 2 \sqrt{2} \cdot \sqrt{2} + 2 \sqrt{2}$

$\textcolor{w h i t e}{\text{XXXX}}$$= 2 \cdot 2 + 2 \sqrt{2}$

$\textcolor{w h i t e}{\text{XXXX}}$$= 4 + 2 \sqrt{2}$

Jun 29, 2015

$\sqrt{2} \left(\sqrt{8} + \sqrt{4}\right) = 2 \left(2 + \sqrt{2}\right)$

#### Explanation:

1. $\sqrt{2} \left(\sqrt{8} + \sqrt{4}\right) = \sqrt{2} \cdot \sqrt{8} + \sqrt{2} \cdot \sqrt{4}$ for the distributive property.
2. $\sqrt{2} \cdot \sqrt{8} + \sqrt{2} \cdot \sqrt{4} = \sqrt{2 \cdot 8} + \sqrt{2 \cdot 4} = \sqrt{{2}^{4}} + \sqrt{{2}^{3}}$ because we can write the product of two square roots as the square roots of the product. I rewrote the products in exponential form, it is easier.
3. Now, if you remember, the square root of x means x at the exponent of 1/2: $\sqrt{x} = {x}^{\frac{1}{2}}$.
So in this case we can write:
$\sqrt{{2}^{4}} + \sqrt{{2}^{3}} = {2}^{\frac{4}{2}} + {2}^{\frac{3}{2}}$
4. Now we can solve the equation:
${2}^{\frac{4}{2}} + {2}^{\frac{3}{2}} = {2}^{2} + {2}^{1 + \frac{1}{2}} = 4 + 2 \cdot {2}^{\frac{1}{2}} = 4 + 2 \sqrt{2} = 2 \left(2 + \sqrt{2}\right)$.