Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Simplify. (sqrt5+sqrt3)/(sqrt5-sqrt3)+(sqrt5-sqrt3)/(sqrt5+sqrt3)

Simplify.\newline5+35−3+5−35+3\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Multiply radical expressionsright chevron icon

Full solution

Q. Simplify.\newline5+35−3+5−35+3\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}
  1. Consider Fractions Separately: We have the expression (5+3)/(5−3)+(5−3)/(5+3)(\sqrt{5} + \sqrt{3}) / (\sqrt{5} - \sqrt{3}) + (\sqrt{5} - \sqrt{3}) / (\sqrt{5} + \sqrt{3}). To simplify, we will first consider each fraction separately and then add them together.
  2. Multiply by Conjugate: To simplify the fractions, we can multiply the numerator and denominator of each fraction by the conjugate of the denominator. The conjugate of (5−3)(\sqrt{5} - \sqrt{3}) is (5+3)(\sqrt{5} + \sqrt{3}), and the conjugate of (5+3)(\sqrt{5} + \sqrt{3}) is (5−3)(\sqrt{5} - \sqrt{3}).
  3. Simplify First Fraction: For the first fraction, multiply the numerator and denominator by the conjugate of the denominator: (5+35−3)⋅(5+35+3)\left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right) \cdot \left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right).
  4. Divide Numerator and Denominator: Perform the multiplication in the numerator and denominator:\newlineNumerator: (5+3)×(5+3)=5+215+3(\sqrt{5} + \sqrt{3}) \times (\sqrt{5} + \sqrt{3}) = 5 + 2\sqrt{15} + 3.\newlineDenominator: (5−3)×(5+3)=5−3(\sqrt{5} - \sqrt{3}) \times (\sqrt{5} + \sqrt{3}) = 5 - 3.
  5. Simplify Second Fraction: Simplify the numerator and denominator:\newlineNumerator: 5+215+3=8+2155 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}.\newlineDenominator: 5−3=25 - 3 = 2.\newlineSo the first fraction simplifies to (8+215)/2(8 + 2\sqrt{15}) / 2.
  6. Divide Numerator and Denominator: Divide each term in the numerator by the denominator: (8+215)/2=4+15(8 + 2\sqrt{15}) / 2 = 4 + \sqrt{15}.
  7. Add Simplified Fractions: Now, we repeat the process for the second fraction, multiplying the numerator and denominator by the conjugate of the denominator: (5−35+3)×(5−35−3)\left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right) \times \left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right).
  8. Combine Like Terms: Perform the multiplication in the numerator and denominator:\newlineNumerator: (5−3)×(5−3)=5−215+3(\sqrt{5} - \sqrt{3}) \times (\sqrt{5} - \sqrt{3}) = 5 - 2\sqrt{15} + 3.\newlineDenominator: (5+3)×(5−3)=5−3(\sqrt{5} + \sqrt{3}) \times (\sqrt{5} - \sqrt{3}) = 5 - 3.
  9. Final Simplified Form: Simplify the numerator and denominator:\newlineNumerator: 5−215+3=8−2155 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}.\newlineDenominator: 5−3=25 - 3 = 2.\newlineSo the second fraction simplifies to (8−215)/2(8 - 2\sqrt{15}) / 2.
  10. Final Simplified Form: Simplify the numerator and denominator:\newlineNumerator: 5−215+3=8−2155 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}.\newlineDenominator: 5−3=25 - 3 = 2.\newlineSo the second fraction simplifies to (8−215)/2(8 - 2\sqrt{15}) / 2.Divide each term in the numerator by the denominator:\newline(8−215)/2=4−15(8 - 2\sqrt{15}) / 2 = 4 - \sqrt{15}.
  11. Final Simplified Form: Simplify the numerator and denominator:\newlineNumerator: 5−215+3=8−2155 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}.\newlineDenominator: 5−3=25 - 3 = 2.\newlineSo the second fraction simplifies to (8−215)/2(8 - 2\sqrt{15}) / 2.Divide each term in the numerator by the denominator:\newline(8−215)/2=4−15(8 - 2\sqrt{15}) / 2 = 4 - \sqrt{15}.Now, add the simplified forms of the two fractions together:\newline(4+15)+(4−15)(4 + \sqrt{15}) + (4 - \sqrt{15}).
  12. Final Simplified Form: Simplify the numerator and denominator:\newlineNumerator: 5−215+3=8−2155 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}.\newlineDenominator: 5−3=25 - 3 = 2.\newlineSo the second fraction simplifies to (8−215)/2(8 - 2\sqrt{15}) / 2.Divide each term in the numerator by the denominator:\newline(8−215)/2=4−15(8 - 2\sqrt{15}) / 2 = 4 - \sqrt{15}.Now, add the simplified forms of the two fractions together:\newline(4+15)+(4−15)(4 + \sqrt{15}) + (4 - \sqrt{15}).Combine like terms:\newline4+15+4−15=84 + \sqrt{15} + 4 - \sqrt{15} = 8.
  13. Final Simplified Form: Simplify the numerator and denominator:\newlineNumerator: 5−215+3=8−2155 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}.\newlineDenominator: 5−3=25 - 3 = 2.\newlineSo the second fraction simplifies to (8−215)/2(8 - 2\sqrt{15}) / 2.Divide each term in the numerator by the denominator:\newline(8−215)/2=4−15(8 - 2\sqrt{15}) / 2 = 4 - \sqrt{15}.Now, add the simplified forms of the two fractions together:\newline(4+15)+(4−15)(4 + \sqrt{15}) + (4 - \sqrt{15}).Combine like terms:\newline4+15+4−15=84 + \sqrt{15} + 4 - \sqrt{15} = 8.The final simplified form of the expression is 88.

More problems from Multiply radical expressions

Question
Simplify. \newline925\sqrt{\frac{9}{25}}\newline_____________
Get tutor helpright-arrow

Posted 5 months ago

Question
Multiply. Write your answer in simplest form.\newline15×2\sqrt{15} \times \sqrt{2}\newline_____________
Get tutor helpright-arrow

Posted 5 months ago

Question
Simplify. \newline125 \sqrt{125}
Get tutor helpright-arrow

Posted 7 months ago

Question
Add. Write your answer in simplest form.\newline−63+73-6\sqrt{3} + 7\sqrt{3}\newline______
Get tutor helpright-arrow

Posted 5 months ago

Question
Multiply. Write your answer in simplest form. \newline−2(5−5)-\sqrt{2}(5 - \sqrt{5})
Get tutor helpright-arrow

Posted 5 months ago

Question
Simplify. Rationalize the denominator.\newline`(-7)/(2 + \sqrt{5}\)`\newline______
Get tutor helpright-arrow

Posted 5 months ago

Question
Simplify. Assume x x is greater than or equal to zero.\newline20x3 \sqrt{20x^3}
Get tutor helpright-arrow

Posted 5 months ago

Question
4981=\sqrt{\frac{49}{81}}=
Get tutor helpright-arrow

Posted 3 months ago

Question
5123=\sqrt[3]{512}=
Get tutor helpright-arrow

Posted 2 months ago

Question
10003=\sqrt[3]{1000}=
Get tutor helpright-arrow

Posted 2 months ago

View all (26)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant