# SIMPLIFYING SQUARE ROOTS

Simplifying Square Roots :

In this section, we are going to learn how to simplify square roots.

Thew following steps will be useful to simplify any square root.

(i)  Decompose the number inside the square root into prime factors.

(ii)  Inside the square root, if the same number is repeated twice with multiplication, it can be taken out of the square root.

(iii) Combine the like square root terms using mathematical operations.

Example :

√27 + √9 - √12  =  √(3 ⋅ 3 ⋅ 3) + (3 ⋅ 3) - √(2 ⋅ 2 ⋅ 3)

√27 + √9 - √12  =  3√3 + √3 - 2√3

√27 + √9 - √12  =  2√3

## Simplifying Square Roots - Examples

Example 1 :

Simplify the following square root expression :

√64 + √196

Solution :

Decompose 64 and 196 into prime factors using synthetic division.

 √64  =  √(8 ⋅ 8)√64  =  8 √196  =  √(14 ⋅ 14)√196  =  14

So, we have

√64 + √196  =  8 + 14

√64 + √196  =  22

Example 2 :

Simplify the following square root expression :

√40 + √160

Solution :

Decompose 40 and 160 into prime factors using synthetic division.

√40  =  √(2 ⋅ 2 ⋅ 2 ⋅ 5)  =  2√10

√160  =  √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5)  =  4√10

So, we have

√40 + √160  =  2√10 + 4√10

√40 + √160  =  6√10

Example 3 :

Simplify the following square root expression :

2√425 - 3√68

Solution :

Decompose 425 and 68 into prime factors using synthetic division.

 √425  =  √(5 ⋅ 5 ⋅ 17)√425  =  5√17 √68  =  √(2 ⋅ 2 ⋅ 17)√68  =  2√17

So, we have

2√425 - 3√68  =  2(5√17) - 3(2√17)

2√425 - 3√68  =  10√17 - 6√17

2√425 - 3√68  =  4√17

Example 4 :

Simplify the following square root 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 following square root expression :

-√117 - √52

Solution :

Decompose 117 and 52 into prime factors using synthetic division.

√117  =  √(3 ⋅ 3 ⋅ 13)  =  3√13

√52  =  √(2 ⋅ 2 ⋅ 13)  =  2√13

So, we have

-√117 - √52  =  -3√13 - 2√13

-√117 - √52  =  -5√13

Example 6 :

Simplify the following square root expression :

(√17)(√51)

Solution :

Decompose 17 and 51 into prime factors.

Because 17 is a prime number, it can't be decomposed anymore. So, √17 has to be kept as it is.

√51  =  √(3 ⋅ 17)  =  √3 ⋅ √17

So, we have

(√17)(√51)  =  (√17)(√3 ⋅ √17)

(√17)(√51)  =  (√17 ⋅ √17)√3

(√17)(√51)  =  17√3

Example 7 :

Simplify the following square root expression :

(√35)(2√15)

Solution :

Decompose 35 and 15 into prime factors.

√35  =  √(5 ⋅ 7)  =  √5 ⋅ √7

√15  =  √(5 ⋅ 3)  =  √5 ⋅ √3

So, we have

(√35)(2√15)  =  ( √5 ⋅ √7⋅ 2(√5 ⋅ √3)

(√35)(2√15)  =  2(√5 ⋅ √5)(√7 ⋅ √3)

(√35)(2√15)  =  2(5)(√(7 ⋅ 3)

(√35)(2√15)  =  10√21

Example 8 :

Simplify the following square root expression :

(14√117) ÷ (7√52)

Solution :

Decompose 117 and 52 into prime factors using synthetic division.

 √117  =  √(3 ⋅ 3 ⋅ 13)√117  =  3√13 √52  =  √(2 ⋅ 2 ⋅ 13)√52  =  2√13

(14√117) ÷ (7√52)  =  14(3√13) ÷ 7(2√13)

(14√117) ÷ (7√52)  =  42√13 ÷ 14√13

(14√117) ÷ (7√52)  =  42√13 / 14√13

(14√117) ÷ (7√52)  =  3

Example 9 :

Simplify the following square root expression :

(7√5)2

Solution :

(7√5)=  7√5 ⋅ 7√5

(7√5)2  =  (7 ⋅ 7)(√5 ⋅ √5)

(7√5)2  =  (49)(5)

(7√5)2  =  245

Example 10 :

Simplify the following square root expression :

(√3)3 + √27

Solution :

(√3)3 + √27  =  (√3 ⋅ √3  √3) + √(3 ⋅ 3 ⋅ 3)

(√3)3 + √27  =  (3  √3) + 3√3

(√3)3 + √27  =  3√3 + 3√3

(√3)3 + √27  =  6√3

After having gone through the stuff given above, we hope that the students would have understood, "Simplifying Square Roots".

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

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear 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 fractrions

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