Radical Equation

√(5w+13) = √(7w+4) 

just square both sides to get rid of the square root signs 

5w + 13 = 7w + 4 {squared both sides} 
5w = 7w - 9 {subtracted 13 from both sides}
-2w = -9 {subtracted 7w from both sides}
w = 9/2 {divided both sides by -2}

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Simplify the Radical Expression

3√-12 + 4√-48
= 3√-1 √4 √3 + 4√-1 √16 √3 {broke down square roots}
= 3i(2)√3 + 4i(4)√3 {evaluated square roots, √-1 is i}
= 6i√3 + 16i√3 {multiplied}
= 20i√3 {combined like terms}

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Foil Method with Radicals

(9 + √10)(9 - √10)

Use the foil method when multiplying two binomials.

 = 9(9) + 9(-√10) + √10(9) + √10(-√10) {used foil method}
= 81 - 9√10 + 9√10 - √100 {multiplied through}
= 81 - √100 {combined like terms}
= 81 - 10 {evaluated square root of 100 to be 10}
= 71 {subtracted}

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Simplify by rationalizing the denominator

  4
--------
2 - √3  

{multiply top and bottom by conjugate of denominator, (2 + √3)}

   4 (2 + √3)
------------------ 
(2 - √3)(2 + √3) 

{use distributive property on top and foil method on bottom}

8 + 4√3
------------ 
  4 - √9 

 8 + 4√3
----------- {evaluated √9 to be 3 on bottom}
  4 - 3 

= 8 + 4√3 {subtracted 4 - 3 on bottom}

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Simplify the Radical Expression

35√3
----
3√5 

Multiply top and bottom by √5 to eliminate square root sign in denominator

35√3 (√5)
------------
3√5 (√5) 

35√15
-------- {multiplied}
3√25 

35√15
-------- {square root of 25 is 5}
3(5) 
35√15
-------- {multiplied 3 by 5}
15 

7√15
------ {cancelled 35 and 15}
3 

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Simplify the Radical Expression
 
√180 - √120
 
Break down 180 and 120, each into two factors, one of which has a perfect square root

= √36 √5 - √4 √30 {broke down 180 and 120}
= 6√5 - 2√30 {evaluated square roots o f 36 and 4}

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Multiply the radicals

√6 √30
= √180 {multiplied together}
  = √36 √5 {broke square root of 180 down}
  = 6√5 {evaluated the square root of 36 to be 6} 

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