We have an answer to "What is the simplified form of 2−102+10 ?"

High School
Algebra
Algebra I
Equations and inequalities

Nathaly Rivers

Answered question

2022-12-30

What is the simplified form of

\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}}

Answer & Explanation

Collin Johns

Beginner2022-12-31Added 10 answers

The identity of the square that differs is written as follows:

a^{2} - b^{2} = (a - b) (a + b)

Therefore, by multiplying both the numerator and the denominator by, we can rationalize the denominator of the given expression. #sqrt(2)-sqrt(10)# as follows:

\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}} = \frac{{(\sqrt{2} - \sqrt{10})}^{2}}{(\sqrt{2} - \sqrt{10}) (\sqrt{2} + \sqrt{10})}

\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}} = \frac{{(\sqrt{2})}^{2} - 2 \sqrt{2} \sqrt{10} + {(\sqrt{10})}^{2}}{{(\sqrt{2})}^{2} - {(\sqrt{10})}^{2}}

\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}} = \frac{2 - 2 \sqrt{20} + 10}{2 - 10}

\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}} = \frac{12 - 2 \sqrt{2^{2} \cdot 5}}{- 8}

\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}} = \frac{12 - 4 \sqrt{5}}{- 8}

\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}} = - \frac{3}{2} + \frac{1}{2} \sqrt{5}

Exceplyclene72

Beginner2023-01-01Added 1 answers

Given

\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}}

Let's rationalise the denominator:

= \frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}} \cdot \frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} - \sqrt{10}}

= \frac{{(\sqrt{2} + \sqrt{10})}^{2}}{{(\sqrt{2})}^{2} - {(\sqrt{10})}^{2}}

= \frac{2 - 2 \sqrt{20} + 10}{2 - 10}

= \frac{12 - 2 \sqrt{2^{2} \cdot 5}}{- 8}

= \frac{12 - 2 \cdot 2 \sqrt{5}}{- 8}

= \frac{12 - 4 \sqrt{5}}{- 8}

= - \frac{6 - 2 \sqrt{5}}{4}

= - \frac{3 - \sqrt{5}}{2}

= - \frac{3}{2} + \frac{1}{2} \sqrt{5}

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