How do you simplify sqrt (16) / (sqrt (4) + sqrt (2))?

It is

sqrt (16) / (sqrt (4) + sqrt (2))=4/[sqrt2(sqrt2+1)]= 2sqrt2/(sqrt2+1)=2*sqrt2(sqrt2-1)/[(sqrt2+1)*(sqrt2-1)]= 2sqrt2(sqrt2-1)

Jun 26, 2016

$4 - 2 \sqrt{2}$

Explanation:

Try to rationalize the denominator. Multiply numerator and denominatr by $\left(\sqrt{4} - \sqrt{2}\right)$

$\sqrt{16} \frac{\sqrt{4} - \sqrt{2}}{\left(\sqrt{4} + \sqrt{2}\right) \cdot \left(\sqrt{4} - \sqrt{2}\right)}$

$4 \cdot \frac{2 - \sqrt{2}}{4 - 2}$

$4 \cdot \frac{2 - \sqrt{2}}{2}$

$2 \cdot \left(2 - \sqrt{2}\right)$

$4 - 2 \sqrt{2}$

Multiply through by $\frac{2 - \sqrt{2}}{2 - \sqrt{2}}$ and work through to get $4 - 2 \sqrt{2} = 2 \left(2 - \sqrt{2}\right)$

Explanation:

$\frac{\sqrt{16}}{\sqrt{4} + \sqrt{2}}$

Let's first take the square roots of the perfect squares:

$\frac{4}{2 + \sqrt{2}}$

In order to simplify, we need the square root out from the denominator. The way to do this is to ensure that when we do FOIL (the process of multiplying 2 quantities within brackets), we don't end up with more square roots. To do that, we'll multiply by $\left(2 - \sqrt{2}\right)$ which will eliminate that possibility (like this):

$\frac{4}{2 + \sqrt{2}} \cdot \left(\frac{2 - \sqrt{2}}{2 - \sqrt{2}}\right)$

$\frac{4 \cdot 2 - 4 \sqrt{2}}{2 \cdot 2 - 2 \sqrt{2} + 2 \sqrt{2} - \sqrt{2} \sqrt{2}}$

$\frac{8 - 4 \sqrt{2}}{4 - 2} = \frac{8 - 4 \sqrt{2}}{2} = 4 - 2 \sqrt{2} = 2 \left(2 - \sqrt{2}\right)$
The text contains a mathematical question and provides a detailed answer to it, including multiple steps of simplification and rationalization.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
How do you simplify sqrt (16) / (sqrt (4) + sqrt (2))?

It is

sqrt (16) / (sqrt (4) + sqrt (2))=4/[sqrt2(sqrt2+1)]= 2sqrt2/(sqrt2+1)=2*sqrt2(sqrt2-1)/[(sqrt2+1)*(sqrt2-1)]= 2sqrt2(sqrt2-1)

Jun 26, 2016

$4 - 2 \sqrt{2}$

Explanation:

Try to rationalize the denominator. Multiply numerator and denominatr by $\left(\sqrt{4} - \sqrt{2}\right)$

$\sqrt{16} \frac{\sqrt{4} - \sqrt{2}}{\left(\sqrt{4} + \sqrt{2}\right) \cdot \left(\sqrt{4} - \sqrt{2}\right)}$

$4 \cdot \frac{2 - \sqrt{2}}{4 - 2}$

$4 \cdot \frac{2 - \sqrt{2}}{2}$

$2 \cdot \left(2 - \sqrt{2}\right)$

$4 - 2 \sqrt{2}$

Multiply through by $\frac{2 - \sqrt{2}}{2 - \sqrt{2}}$ and work through to get $4 - 2 \sqrt{2} = 2 \left(2 - \sqrt{2}\right)$

Explanation:

$\frac{\sqrt{16}}{\sqrt{4} + \sqrt{2}}$

Let's first take the square roots of the perfect squares:

$\frac{4}{2 + \sqrt{2}}$

In order to simplify, we need the square root out from the denominator. The way to do this is to ensure that when we do FOIL (the process of multiplying 2 quantities within brackets), we don't end up with more square roots. To do that, we'll multiply by $\left(2 - \sqrt{2}\right)$ which will eliminate that possibility (like this):

$\frac{4}{2 + \sqrt{2}} \cdot \left(\frac{2 - \sqrt{2}}{2 - \sqrt{2}}\right)$

$\frac{4 \cdot 2 - 4 \sqrt{2}}{2 \cdot 2 - 2 \sqrt{2} + 2 \sqrt{2} - \sqrt{2} \sqrt{2}}$

$\frac{8 - 4 \sqrt{2}}{4 - 2} = \frac{8 - 4 \sqrt{2}}{2} = 4 - 2 \sqrt{2} = 2 \left(2 - \sqrt{2}\right)$
user
The text contains a mathematical question and provides a detailed answer to it, including multiple steps of simplification and rationalization.

TYPE: SINGLE_QUESTION_WITH_ANSWER