SIMPLIFYING RADICAL EXPRESSIONS

The following steps will be useful to simplify any radical expressions. 

Step 1 :

Decompose the number inside the radical into prime factors. 

Step 2 :

If you have square root (√), you have to take one term out of the square root for every two same terms multiplied inside the radical. 

Step 3 :

If you have cube root (³√), you have to take one term out of cube root for every three same terms multiplied inside the radical.

Step 4 :

If you have fourth root (⁴√), you have to take one term out of fourth root for every four same terms multiplied inside the radical.

Step 5 :

Combine the radical terms using mathematical operations. 

Example : 

√18 + √8  =  √(3 ⋅ 3 ⋅ 2) + √(2 ⋅ 2 ⋅ 2)

 √18 + √8  =  3√2 + 2√2

 √18 + √8  =  5√2

Solved Examples 

Example 1 : 

Simplify the radical expression : 

√169 + √121

Solution : 

Decompose 169 and 121 into prime factors using synthetic division. 

So, we have

√169 + √121  =  13 + 11

√169 + √121  =  24

Example 2 : 

Simplify the radical expression : 

√20 + √320

Solution : 

Decompose 20 and 320 into prime factors using synthetic division. 

So, we have

√20 + √320  =  2√5 + 8√5

√20 + √320  =  10√5

Example 3 : 

Simplify the radical expression : 

√117 - √52

Solution : 

Decompose 117 and 52 into prime factors using synthetic division. 

So, we have

√117 - √52  =  3√13 - 2√13

√117 + √52  =  √13

Example 4 : 

Simplify the radical expression : 

√243 - 5√12 + √27 

Solution : 

Decompose 243, 12 and 27 into prime factors using synthetic division. 

√243  =  √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)  =  9√3

√12  =  √(2 ⋅ 2 ⋅ 3)  =  2√3

√27  =  √(3 ⋅ 3 ⋅ 3)  =  3√3

So, we have

√243 - 5√12 + √27  =  9√3 - 5(2√3) + 3√3

Simplify.

√243 - 5√12 + √27  =  9√3 - 10√3 + 3√3

√243 - 5√12 + √27  =  2√3

Example 5 : 

Simplify the radical expression : 

-√147 - √243 

Solution : 

Decompose 147 and 243 into prime factors using synthetic division. 

√147  =  √(7 ⋅ 7 ⋅ 3)  =  7√3

√243  =  √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)  =  9√3

So, we have

-√147 - √243  =  -7√3 - 9√3

-√147 - √243  =  -16√3

Example 6 : 

Simplify the radical expression : 

(√13)(√26)

Solution : 

Decompose 13 and 26 into prime factors. 

13 is a prime number. So, it can't be decomposed anymore.

√26  =  √(2 ⋅ 13)  =  √2 ⋅ √13

So, we have

(√13)(√26)  =  (√13)(√2 ⋅ √13)

(√13)(√26)  =  (√13 ⋅ √13)√2

(√13)(√26)  =  13√2

Example 7 : 

Simplify the radical expression : 

(3√14)(√35)

Solution : 

Decompose 14 and 35 into prime factors.

√14  =  √(2 ⋅ 7)  =  √2 ⋅ √7

√35  =  √(5 ⋅ 7)  =  √5 ⋅ √7

So, we have

(3√14)(√35)  =  3( √2 ⋅ √7)(√5 ⋅ √7)

(3√14)(√35)  =  3(√7 ⋅ √7)(√2 ⋅ √5)

(3√14)(√35)  =  3(7)√(2 ⋅ 5)

(3√14)(√35)  =  21√10

Example 8 : 

Simplify the radical expression : 

(8√117) ÷ (2√52)

Solution : 

Decompose 117 and 52 into prime factors using synthetic division.

(8√117) ÷ (2√52)  =  8(3√13) ÷ 2(2√13)

(8√117) ÷ (2√52)  =  24√13 ÷ 4√13

(8√117) ÷ (2√52)  =  24√13 / 4√13

(8√117) ÷ (2√52)  =  6

Example 9 : 

Simplify the radical expression : 

(8√3)²

Solution :

(8√3)²  =  8√3 ⋅ 8√3

(8√3)²  =  (8 ⋅ 8)(√3 ⋅ √3)

(8√3)²  =  (64)(3)

(8√3)²  =  192

Example 10 : 

Simplify the radical expression : 

(√2)³ + √8

  

Solution :

(√2)³ + √8  =  (√2 ⋅ √2 ⋅ √2) + √(2⋅ 2 ⋅ 2)

(√2)³ + √8  =  (2 ⋅ √2) + 2√2

(√2)³ + √8  =  2√2 + 2√2

(√2)³ + √8  =  4√2

Example 11 :

Simplify : 

⁴√(x⁴/16)

Solution :

⁴√(x⁴/16)  =  ⁴√(x⁴) / ⁴√16

⁴√(x⁴/16)  =  ⁴√(x ⋅ x ⋅ x ⋅ x) / ⁴√(2 ⋅ 2 ⋅ 2 ⋅ 2)

⁴√(x⁴/16)  =  x / 2

Example 12 :

Simplify : 

³√(125p⁶q³)

Solution :

³√(125p⁶q³)  =  ³√(5 ⋅ 5 ⋅ 5 ⋅ p² ⋅ p² ⋅ p² ⋅ q ⋅ q ⋅ q)

³√(125p⁶q³)  =  5p²q

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and Venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6