# How do you simplify  sqrt ((4a^3 )/( 27b^3))?

Jun 9, 2016

$\sqrt{\frac{4 {a}^{3}}{27 {b}^{3}}} = \frac{2 a}{3 b} \sqrt{\frac{a}{3 b}}$

#### Explanation:

$\sqrt{\frac{4 {a}^{3}}{27 {b}^{3}}}$

= $\sqrt{\frac{2 \times 2 \times a \times a \times a}{3 \times 3 \times 3 \times b \times b \times b}}$

= $\sqrt{\frac{\underline{2 \times 2} \times \underline{a \times a} \times a}{\underline{3 \times 3} \times 3 \times \underline{b \times b} \times b}}$

= $\frac{2 a}{3 b} \sqrt{\frac{a}{3 \times b}}$

= $\frac{2 a}{3 b} \sqrt{\frac{a}{3 b}}$

Jun 10, 2016

Just another way of writing the same thing as the other solution:

$\text{ } \textcolor{g r e e n}{\frac{2 a \sqrt{3 a b}}{3 {b}^{2}}}$

#### Explanation:

Given:$\text{ } \sqrt{\frac{4 {a}^{3}}{27 {b}^{3}}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Concept of approach}}$

You look for numbers that are squared. Take them outside the square root and 'get rid' of the square.

Example: Suppose we had $\sqrt{12}$

Choosing the factors of $3 \times 4 = 12$ we write it as:

$\sqrt{3 \times 4}$ but 4 is ${2}^{2}$ giving $\sqrt{3 \times {2}^{2}}$

Take the 2 out side the square root giving

$\text{ } 2 \sqrt{3}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Solving your question}}$

$\textcolor{b r o w n}{\text{Mathematicians do not like square roots in the denominator so we will 'get rid' of it}}$$\textcolor{b r o w n}{\text{later.}}$

$\text{ Write as: } \frac{\sqrt{4 {a}^{3}}}{\sqrt{27 {b}^{3}}}$
'.......................................................................................
we know that
$27 \text{ "=" "3xx9" " =" "3xx3^2" and "b^3" "=" } {b}^{2} \times b$
$4 \text{ "=2^2" and " a^3" } = {a}^{2} \times a$
'.........................................................................................

$\text{ Write as } \frac{\sqrt{{2}^{2} \times {a}^{2} \times a}}{\sqrt{{3}^{2} \times 3 \times {b}^{2} \times b}}$

Take the squared values outside the square roots giving

$\text{ } \textcolor{b l u e}{\frac{2 a \sqrt{a}}{3 b \sqrt{3 b}}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Now to 'get rid' of the square root in the denominator}}$

Multiply by 1 and you do not change the value or how it looks.
1 can come in many forms: $\frac{1}{1} \text{; "(-1)/(-1)"; "2/2"; } \frac{\sqrt{3 b}}{\sqrt{3 b}}$

So we can multiply by 1 and not change the inherent value but we can change the way it looks.

$\textcolor{b r o w n}{\text{Multiply by 1 but in the form of } 1 = \frac{\sqrt{3 b}}{\sqrt{3 b}}}$

$\frac{2 a \sqrt{a}}{3 b \sqrt{3 b}} \times \frac{\sqrt{3 b}}{\sqrt{3 b}} \text{ "=" } \frac{2 a \sqrt{a} \sqrt{3 b}}{3 b \sqrt{3 b} \sqrt{3 b}}$

But $\sqrt{3 b} \times \sqrt{3 b} = 3 b$ giving:

$\frac{2 a \sqrt{a} \sqrt{3 b}}{3 b \times b} \text{ "=" } \frac{2 a \sqrt{a} \sqrt{3 b}}{3 {b}^{2}}$

But $\sqrt{a} \sqrt{3 b} = \sqrt{3 a b}$

$\textcolor{b r o w n}{\text{If you can take things out of square roots you can put things back in!}}$

$\text{ } \textcolor{g r e e n}{\frac{2 a \sqrt{3 a b}}{3 {b}^{2}}}$