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Surds work book

Write your name in this box

__________________________________________________________________________

How to use this workbook

Where to write your answersYou will see on each question there are one or more grey boxes. These are for you to type the answer in.

Eg: 6 + 4 =

In the box you write your answer.

The layout keeps changingOne problem is that the grey boxes are a different size when empty, to when full. This means that questions take up a different amount of space when empty. You will see this means the pages will change as you fill them in. Unfortunately there is nothing I can do about that.

Why are you asking me questions about my answers? You will also see that on some questions you are asked to write what has changed from one question to the next. Most questions are in a specific order and only one thing has changed from one question to the next. By looking at what changes from one question to the next, you can see what affect that change has on the answer. That should help you to understand what is going on.

You are also asked to say what the affect this had on the answer. Did it double it? Did it increase it by 3, reduce it by a power of x? Once again, by looking at what impact the change had on the answer, it will help you understand what is going on in the question.

Squares

12 = 122 = 432 = 942 = 1652 = 25

62 = 3672 = 4982 = 6492 = 81102 = 100

112 = 121122 = 144132 = 169142 = 196152 = 225

Rules of surds

√a×b=√a×√b

√ ab=√a√b

√a×√a=a

BUT √a+b≠√a+√b

Worksheet 1: Products of squares and integers

Example 1

Factorise these integers into a square number and an integer75 = 3 × 25

= 3 × 52

1. 12 = 3 x 4

= 3 x 2

2. 20 = 5 x 4

= 5 x 2

3. 45 = 5 x

= 5 x 2

4. 50 = x

= x 2

5. 27 = x

= x 2

6. 32 = x

= x 2

7. 108 = x

= x 2

8. 300 = x

= x 2

9. 98 = x

= x 2

10. 192 = x

= x 2

11. 147 = x

= x 2

12. 245 = x

= x 2

13. 243 = x

= x 2

14. 48 = x

= x 2

15. 363 = x

= x 2

16. 605 = x

= x 2

17. 180 = x

= x 2

18. 252 = x

= x 2

19. 288 = x

= x 2

20. 200 = x

= x 2

Worksheet 2: What is a surd?

A surd is an irrational number, that is a number that cannot be written in the form pq where

p and q are both integers. Numbers that can be written in the form pq where p and q are

both integers are called rational. Note: All integers are rational as you can write them with a

denominator of 1. e.g. 5 = 51

Example 1

√4 is not a surd (√4 = 2, which can be written as 21)

√2 is a surd. (√2 is irrational)2√2 is a surd. (√2 is irrational, so 2 × √2 is also irrational)2√4 is not a surd (√4 = 2, so 2 × √4 is 2 × 2, which is rational)

√64 is not a surd (√64 = 8, which can be written as 81 ).

√ 6425 is NOT a surd as it can be written as √64√25 = 85 .

Another way of thinking of this, is the number a square number? The square roots of square numbers are rational.

Are the following surds? Write Y or N in the box

√1 X√4 X√9 X√25

X√20 X√49 X√40

X3√2 X2√1 X2√9

X3√9 X4 √9 X3√3

X3√27 X7√25 X9√2

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Worksheet 3: simplifying surdsWhen we simplify surds, we want to write them with the smallest possible root multiplied by an integer. Remember: √a×b=√a×√b

Example 1Simplify √12We can write 12 as the product of a square number and an integer 12 = 4 × 3√12=√4×3

¿√4×√3 ¿√22×√3 ¿2×√3¿2√3

Example 2Simplify √700700 = 100 × 7√700=√100×7

¿√100×√7 ¿√102×√7 ¿10×√7¿10√7

1. √8 = √4×2= √4×√2= 2√FORMTEXT

2. √32 = √16×2= √16×√FORMTEXT= 4√FORMTEXT

What changed between Q1 and Q2 How did this affect the answer?

3. √128 = √64×2=

√FORMTEXT×√FORMTEXT= 8√FORMTEXT


4. √192 = √64×3=

√FORMTEXT×√FORMTEXT= √FORMTEXT

What changed between Q3and Q4 How did this affect the answer?

5. √320 = √64×FORMTEXT=



6. √80 = √FORMTEXT× FORMTEXT

= √FORMTEXT×√FORMTEXT

= √FORMTEXT




= √FORMTEXT




= √FORMTEXT




= √FORMTEXT




=




= √FORMTEXT




= √FORMTEXT




= √FORMTEXT




= √FORMTEXT


Worksheet 4: simplifying surds- IIMore advanced surd expressions. Remember, our aim is to have the smallest possible root. First write the surd as the product of a square number and an integer.

Example 1Simplify the surd 4 √8We can write √8 as √4×2=√4×√2Which is the same as 2√2So our calculation is now 4×2×√2 which is 8√2

Example 2Simplify the surd 4 √72We can write √72 as √36×2=√36×√2Which is the same as 6√2So our calculation is now 4×6×√2 which is 24√2

1. 2√8 = 2×√4×2= 2×√4×√2= 2×2×√2= √2

2. 3√8 = 3×√4×2= 3×√4×√2= 3×FORMTEXT ×√2= √2


3. 5√72 = 5×√36×2=

5×√FORMTEXT ×√FORMTEXT=

5×FORMTEXT ×√FORMTEXT= √2


4. 7√72 = 7×√FORMTEXT ×FORMTEXT

= 7×√FORMTEXT ×√FORMTEXT

= 7×FORMTEXT ×√FORMTEXT= √FORMTEXT


5. 5√1440 = FORMTEXT ×√FORMTEXT ×FORMTEXT

= FORMTEXT ×√FORMTEXT ×√FORMTEXT

= FORMTEXT ×FORMTEXT ×√FORMTEXT

= √FORMTEXT





= √FORMTEXTWhat changed between Q5 and Q6 How did this affect the answer?

7. 5√36 = FORMTEXT ×√FORMTEXT

= FORMTEXT ×FORMTEXT

= What changed between Q6 and Q7 How did this affect the answer?

8. 7√49 = FORMTEXT ×√FORMTEXT

= FORMTEXT ×FORMTEXT

= What changed between Q7 and Q8 How did this affect the answer?

















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Worksheet 5: Multiplying surds

Remember: √a×b=√a×√b

Multiplying surdsExample 1Find √3×√12. Simplify your answerUse √a×√b=√a×b. √3×√12=√3×12=√36 √36=6

Example 2Find √5×√15. Simplify your answerUse √a×√b=√a×b. √5×√15=√5×15=√75 √75=√25×√3=5√3

1. √6×√3 = √6×3= √18= √9×2= √9×√2= √2

2. √24×√3 = √24×3= √72= √FORMTEXT×2= √FORMTEXT×√2= √2

3. √2×√10 = √2×10= √FORMTEXT= √FORMTEXT×5=


4. √3×√15 = √FORMTEXT× FORMTEXT

= √FORMTEXT=

√FORMTEXT× FORMTEXT=



= √FORMTEXT=




= √FORMTEXT=




= √FORMTEXT=




= √FORMTEXT=




= √FORMTEXT=




= √FORMTEXT=



Worksheet 6: Dividing surds

Your answer may be an integer, or it may be in surd form

Remember: √ ab=√a√b

Dividing surdsExample 1Find √40÷√10. Simplify your answer

Use √ ab=√a√b

√40÷√10=√ 4010=√4

√4=2

Example 2Find √80÷√10. Simplify your answer

Use √ ab=√a√b

√80÷√10=√ 8010=√8

√8=√4×2=√4×√2=2√2

1. √90÷√10 ¿√ 9010= √9

= FORMTEXT

2. √72÷√2 ¿√ 722= √FORMTEXT

= FORMTEXT

3. √200÷√8 ¿√ FORMTEXT8

= √FORMTEXT

= FORMTEXT

4. √243÷√3 ¿√ FORMTEXTFORMTEXT

= √FORMTEXT

= FORMTEXT

5. √294÷√6 ¿√ FORMTEXTFORMTEXT

= √FORMTEXT

= FORMTEXT

6. √120÷√10 ¿√ 12010= √12= √4×3¿√4×√3¿ FORMTEXT √3

7. √180÷√3 ¿√ 1803= √60= √4×15

¿√FORMTEXT×√FORMTEXT

¿ FORMTEXT √FORMTEXT

8. √180÷√9 ¿√ 1809= √FORMTEXT=

√FORMTEXT× FORMTEXT



9. √3850÷√22 ¿√ FORMTEXTFORMTEXT= √FORMTEXT=




10. √2520÷√35 ¿√ FORMTEXTFORMTEXT= √FORMTEXT=




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Worksheet 7: Adding and subtracting surds

You can simplify expressions containing surds by collecting like terms.You might have to simplify individual terms first to make the surd parts match.When adding or subtracting, you can only simplify if the surd parts match, so for example you can do the following: 3√5+4 √5=7√5, because they are both multiples of √5.But you cannot add 3√5+3√6 because the surd parts do not match.

Example 1: Addition: Same surd componentSimplify 3√3+2√3¿5√3

Example 2: Subtraction: Same surd componentSimplify 3√3−2√3¿√3

Example 3: Addition: Different surd componentSimplify 2√27+√12√12=√4×√3=2√3

2√27=2×√27¿2×√9×3¿2×√9×√3¿2×3×√3¿6√36√3+2√3¿8√3

Example 4: Subtraction: Different surd componentSimplify 2√27−√12√12=√4×√3=2√3

2√27=2×√27¿2×√9×3¿2×√9×√3¿2×3×√3¿6√36√3−2√3¿4 √3

1. 7√7−3 √7=FORMTEXT √7 2. 7√7+3√7=FORMTEXT √FORMTEXT

3. 3√7−7 √7=FORMTEXT √FORMTEXT

4. 2√32+3√2 ¿2×√16×2+3√2¿2×√16×√2+3√2¿2×4×√2+3√2¿8√2+3√2¿ FORMTEXT √2

5. 4√32+3√2 ¿4×√16×2+3√2¿4×√16×√2+3√2¿4×4×√2+3√2¿ FORMTEXT √2+3 √2



6. 4√32−3√2 ¿4×√16×2−3 √2

¿4×√FORMTEXT×√FORMTEXT−3√2

¿4×FORMTEXT ×√FORMTEXT−3√2

¿ FORMTEXT √FORMTEXT−3√2¿ FORMTEXT √FORMTEXT


7. 2√3+5√12 ¿2√3+5×√4×3¿2×√FORMTEXT+5×√FORMTEXT×√FORMTEXT¿2×√FORMTEXT+5×FORMTEXT ×√FORMTEXT¿ FORMTEXT √FORMTEXT 8. 2√3+6√27 ¿ FORMTEXT √FORMTEXT +6×√FORMTEXT× FORMTEXT¿ FORMTEXT ×√FORMTEXT+6×√FORMTEXT ×√FORMTEXT¿FORMTEXT ×√FORMTEXT+6×FORMTEXT ×√FORMTEXT¿ FORMTEXT √FORMTEXT

9. 6√60+2√240 ¿6×√FORMTEXT ×FORMTEXT+2×√FORMTEXT ×FORMTEXT

¿6×√FORMTEXT ×√FORMTEXT+2×√FORMTEXT ×√FORMTEXT

¿ FORMTEXT ×FORMTEXT ×√FORMTEXT+FORMTEXT ×FORMTEXT ×√FORMTEXT


10. 6√6+2√24 ¿ FORMTEXT ×√FORMTEXT+FORMTEXT ×√FORMTEXT ×FORMTEXT

¿ FORMTEXT ×√FORMTEXT+FORMTEXT ×√FORMTEXT ×√FORMTEXT

¿ FORMTEXT ×√FORMTEXT+FORMTEXT ×FORMTEXT ×√FORMTEXT


11. 6√6+2√24+5√36¿ FORMTEXT ×√FORMTEXT+FORMTEXT ×√FORMTEXT ×FORMTEXT+FORMTEXT ×√FORMTEXT

¿ FORMTEXT ×√FORMTEXT+FORMTEXT ×√FORMTEXT ×√FORMTEXT+FORMTEXT ×FORMTEXT

¿ FORMTEXT ×√FORMTEXT+FORMTEXT ×FORMTEXT ×√FORMTEXT+FORMTEXT

¿ FORMTEXT+FORMTEXT √FORMTEXT


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Worksheet 8: Multiplying brackets using surdsMultiply out brackets with surds in them in exactly the same way as you multiply out brackets involving algebraic variables. Once you have expanded the brackets, simplify the surds if possible.

Example 1: Surd multiplied by a bracket containing a surdExpand and simplify √3 (2+√27 )

2 +√27√3 2√3 √81

¿2√3+√81 .√81=9So: √3 (2+√27 )≡9+2√3

Example 2: Two brackets containing surds multiplied, simplificationExpand and simplify (2−√3 ) (4+√3 )

2 −√34 8 −4√3

+√3 +2√3 −3

8−4 √3+2√3−3¿5−2√3

Example 3: Two brackets containing surds multiplied, no simplificationExpand and simplify (1+√3)(2−√8)

1 +√32 2 +2√3

−√8 −√8 −√24¿2+2√3−√8−√24−√8=−√4×2=−2√2−√24=−√4×6=−2√6None of these surd terms are similar, so we cannot add them. Convention says you list the surds in ascending order:

(1+√3 ) (2−√8 )≡2−2√2+2√3−2√6

1. (√5+2 ¿(√5+1)

√5 +2√5 5 +2√5+1 +√5 +2

¿5+2√5+√5+2¿7+3√FORMTEXT

2. (2+√5)(1+√5)

2 √51 2 +√5

+√5 +2√5 +5

¿ FORMTEXT+FORMTEXT √5+√5+FORMTEXT¿ FORMTEXT+3√FORMTEXT


3. (2−√5)(1+√5)

2 −√51 2 −√5

+√5 +2√5 −5

¿FORMTEXT+FORMTEXT √FORMTEXT−√FORMTEXT−FORMTEXT¿ FORMTEXT+√FORMTEXT


4. (2−√5)(1−√5)

2 −√51 −√FORMTEXT

−√5 −FORMTEXT √FORMTEXT+¿

¿ FORMTEXT−FORMTEXT √FORMTEXT−√FORMTEXT+FORMTEXT¿ FORMTEXT−3√FORMTEXT


5. (√5+2)(√5−1)

√FORMTEXT −FORMTEXT√5 −√FORMTEXT+2 +FORMTEXT √FORMTEXT−¿

¿ FORMTEXT+FORMTEXT √FORMTEXT−√FORMTEXT−FORMTEXT¿FORMTEXT+√FORMTEXT

6. (2+2√5)(1+√5)

2 +2√51 +FORMTEXT √FORMTEXT

+√5 +FORMTEXT √FORMTEXT+¿

¿FORMTEXT+FORMTEXT √FORMTEXT+FORMTEXT √FORMTEXT+FORMTEXT¿FORMTEXT+FORMTEXT √FORMTEXT

7. (√7+2)(√7−1)

√FORMTEXT −FORMTEXT√7 −√FORMTEXT+

+FORMTEXT √FORMTEXT−¿

¿FORMTEXT+FORMTEXT √FORMTEXT−√FORMTEXT−FORMTEXT¿ FORMTEXT+√FORMTEXT

8. (2√7+2)(3√7−1)

FORMTEXT √FORMTEXT−FORMTEXTFORMTEXT √FORMTEXT −FORMTEXT √FORMTEXT

+ +FORMTEXT √FORMTEXT−¿

¿ FORMTEXT+FORMTEXT √FORMTEXT−FORMTEXT √FORMTEXT−FORMTEXT¿ FORMTEXT+FORMTEXT √FORMTEXT

9. (2√7+2)(3√7−6)

FORMTEXT √FORMTEXT−FORMTEXTFORMTEXT √FORMTEXT −FORMTEXT √FORMTEXT

+ +FORMTEXT √FORMTEXT−¿

¿ FORMTEXT+FORMTEXT √FORMTEXT−FORMTEXT √FORMTEXT−FORMTEXT¿ FORMTEXT−FORMTEXT √FORMTEXT


10. (2√7+2)2

FORMTEXT √FORMTEXT+FORMTEXTFORMTEXT √FORMTEXT FORMTEXT √FORMTEXT

+ +FORMTEXT √FORMTEXT+¿

¿FORMTEXT+FORMTEXT √FORMTEXT+FORMTEXT √FORMTEXT+FORMTEXT¿FORMTEXT+FORMTEXT √FORMTEXT

Worksheet 9: Rationalising the denominator - IRationalising the denominator simply means getting rid of the denominator on the bottom. For the simplest expressions, you simply do this by multiplying both top and bottom by the surd value that is the denominator

Example 1: Single surd denominator, integer numerator

Simplify 1√2

Multiply by √2√2 (note √2√2=1 and multiplying anything by 1 leaves it unchanged. This is why

we can do this)

1√2× √2

√2= 1×√2

√2×√2=√22 (remember √2×√2=2)

1√2 simplified = √2

2

Example 2: More complex surd denominator, integer numerator

Simplify 15√5

Multiply by √5√5 (note √5√5=1 and multiplying anything by 1 leaves it unchanged. This is why

we can do this)

15√5

× √5√5

= 1×√55×√5×√5

= √55×5

=√525 (remember √5×√5=5)

15√5 simplified = √5

25

Example 3: More complex surd denominator, surd numerator, no simplification

Simplify √35√5

Multiply by √5√5

√35√5

× √5√5

= √3×√55×√5×√5

= √155×5

=√1525

Example 3: More complex surd denominator, surd numerator, simplification

Simplify √305√5

Multiply by √5√5

√305√5

× √5√5

= √30×√55×√5×√5

=√1505×5

=√25×625

=√25×√625

= 5×√625

=√65

1. 5√2= 5

√2× √2

√2=

¿ 5√2FORMTEXT

2. 6√2= 6

√2× √2

√2=

¿ FORMTEXT √2FORMTEXT

¿ FORMTEXT √2


3. 6√3= 6

√3× √3

√3=

¿ FORMTEXT √FORMTEXTFORMTEXT



4. 4√3= 4

√3× √FORMTEXT

√FORMTEXT =



5. 4√6= FORMTEXT

√FORMTEXT× √FORMTEXT

√FORMTEXT =




6. 52√6

= FORMTEXT2√FORMTEXT

× √FORMTEXT√FORMTEXT

=

¿ FORMTEXT √FORMTEXT12


7. 53√6

= FORMTEXTFORMTEXT √FORMTEXT


=



8. 5

10√6= FORMTEXTFORMTEXT √FORMTEXT


=

¿ √FORMTEXTFORMTEXT


9. 5

10√5= FORMTEXTFORMTEXT √FORMTEXT


=

¿ √FORMTEXTFORMTEXT


10. 102√3

= FORMTEXTFORMTEXT √FORMTEXT


=



Worksheet 10: Difference between two squares

Remember a2−b2=(a−b)(a+b)

Example 1: Factorising simple expressionsx2−9.Is already in the form a2−b2a=x ,b=3x2−9=(x−3)(x+3)

Example 2: Factorising more complex expressions4 x4−9. This is already in the form a2−b2, but recognise that a=2 x2 ((2 x2)2=4 x 4¿∧b=34 x4−9=(2x2−3)(2 x2+3)

Example 3: Factorising more complex expressions - 24 a2 x4−9. This is already in the form a2−b2, but recognise that a=2ax2 ((2ax2 )2=4a2 x4 ¿∧b=34 a2 x4−9 .=(2ax2−3)(2ax2+3)

Example 3: Working out simple calculations(6+3)(6−3) is in the form (a+b ) (a−b ) ,where a=6 ,b=3 (6+4 ) (6−4 )=62−32=36−9=27

Example 4: Working out more complex calculations involving surds(6+√3)(6−√3) is in the form (a+b ) (a−b ) ,where a=6 ,b=√3 (6+√3)(6−√3)62−(3√3 )2=36−3=33

1. x2−16=( x−4 ) ( x+FORMTEXT )

2. 4 x2−16= (2 x−FORMTEXT ) (2 x+FORMTEXT )

3. 4 x4−16=(FORMTEXT x2−FORMTEXT ) (FORMTEXT x2+FORMTEXT )

4. 4 x10−16=(2 xFORMTEXT−FORMTEXT ) (2x FORMTEXT+FORMTEXT )

5. 9 x10−16=(FORMTEXT x FORMTEXT−FORMTEXT ) (FORMTEXT xFORMTEXT+FORMTEXT )

6. 9a2 x10−16=(FORMTEXT axFORMTEXT−FORMTEXT ) (FORMTEXT ax FORMTEXT+FORMTEXT )

7. 81a4 x10−16=(FORMTEXT aFORMTEXT x FORMTEXT−FORMTEXT ) (FORMTEXT aFORMTEXT x FORMTEXT+FORMTEXT )

Worksheet 11: Difference between two squares - II

1. (5+√3)(5−√3) ¿52−(√3 )2

¿25−3¿ FORMTEXT

2. (5−√3)(5+√3) ¿52−(√3 )2

¿ FORMTEXT−FORMTEXT¿ FORMTEXT


3. (7−√3)(7+√3) ¿ FORMTEXT 2−(√3 )2



4. (7−√5)(7+√5) ¿ FORMTEXT 2−(√FORMTEXT )2



5. (5−√7)(5+√7) ¿ FORMTEXT 2−(√FORMTEXT )2



6. (√7−5)(√7+5) ¿ (√FORMTEXT )2−FORMTEXT 2

¿ FORMTEXT−FORMTEXT¿−FORMTEXT


7. (2√7−5)(2√7+5) ¿ (FORMTEXT √FORMTEXT )2−FORMTEXT 2



8. (2√7−12)(2√7+12) ¿ (FORMTEXT √FORMTEXT )2−FORMTEXT 2



9. (12−3 √7)(12+3√7) ¿ FORMTEXT 2−(FORMTEXT √FORMTEXT )2


10. (12−3 √17)(12+3√17) ¿ FORMTEXT2−(FORMTEXT √FORMTEXT )2


Worksheet 12: Rationalising the denominator – forming the conjugate

If the denominator of the surd expression is in the form a±√b, we use the conjugate to give us an integer denominator (i.e. we multiply top and bottom by a∓√b¿If the denominator of the surd expression is in the form √a±b, we use the conjugate to give us an integer denominator (i.e. we multiply top and bottom by √a∓b¿Or put more simply, to form the conjugate, the first term remains the same, the second term, change its sign.

Forming the conjugateThe conjugate of the expression a+√b=a−√bThe conjugate of the expression a−√b=a+√bThe conjugate of the expression √a−b=√a+bThe conjugate of the expression √a+b=√a−b

Example 1Form the conjugate of 3+√2. The conjugate of the expression a+√b=a−√bSo the conjugate of 3+√2 = 3−√2.

Example 2Form the conjugate of 2−√3. The conjugate of the expression a−√b=a+√bSo the conjugate of 2−√3 = 2+√3.

Example 3Form the conjugate of √2+3. The conjugate of the expression √a+b=√a−bSo the conjugate of √2+3 = √2−3.

Example 4Form the conjugate of √2−3. The conjugate of the expression √a−b=√a+bSo the conjugate of √2−3 = √2+3.

Example 5Form the conjugate of −√2−3. The conjugate of the expression √a−b=√a+bSo the conjugate of −√2−3 = −√2+3.

Form the conjugate of:

1. 4+√5 Conjugate = 4−√FORMTEXT 2. 4−√5 Conjugate = FORMTEXT+√FORMTEXT3. √5+4 Conjugate = √FORMTEXT FORMTEXT FORMTEXT4. √5−4 Conjugate = √FORMTEXT FORMTEXT FORMTEXT5. −√5−4 Conjugate = −√FORMTEXT FORMTEXT FORMTEXT

Worksheet 13: Rationalising the denominator – using the conjugate

If the denominator of the surd expression is in the form a±√b, we use the conjugate to give us an integer denominator (i.e. we multiply top and bottom by a∓√b¿If the denominator of the surd expression is in the form √a±b, we use the conjugate to give us an integer denominator (i.e. we multiply top and bottom by √a∓b¿Or put more simply, if the second term is positive, the conjugate has a negative second term. If the second term is negative, the conjugate has a positive second term.

Forming the conjugateThe conjugate of the expression a+√b=a−√bThe conjugate of the expression a−√b=a+√bThe conjugate of the expression √a+b=√a−bThe conjugate of the expression √a−b=√a+b

Example 1: Rationalise the denominator, integer numerator1

2+√2Multiply top and bottom by the conjugate of 2+√21

2+√2× 2−√22−√2

The denominator is the difference between two squares (22−(√2 )2

2−√24−2

=2−√22

Example 2: Rationalise the denominator, surd numerator1+√22+√2Multiply top and bottom by the conjugate of 2+√21+√22+√2

× 2−√22−√2

The denominator is the difference between two squares (22−(√2 )2=4−2=2Expand the numerator

1 +√22 2 +2√2

−√2 −√2 −22+2√2−√2−2¿√2So 1+√22+√2

=√22

1. 23+√2

= 23+√2

× 3−√23−√2 ¿

2×(3−√2)32−√22

¿ 6−2√29−2

¿ 6−2√2FORMTEXT

2. 2

3−√2=¿ 2

3−√2× 3+√23+√2

¿2×(FORMTEXT+√FORMTEXT )

32−√22

¿ FORMTEXT +FORMTEXT √FORMTEXT9−2

¿ FORMTEXT +FORMTEXT √FORMTEXT7


3. 7

3−√2= 73−√2

× FORMTEXT+√FORMTEXTFORMTEXT+√FORMTEXT

¿7×(FORMTEXT+√FORMTEXT )

FORMTEXT 2−√FORMTEXT 2

¿ FORMTEXT +FORMTEXT √FORMTEXTFORMTEXT−FORMTEXT

¿ FORMTEXT +FORMTEXT √FORMTEXTFORMTEXT

¿ FORMTEXT+√FORMTEXT


4. 7

√2−3= 7

√2−3× √FORMTEXT +FORMTEXT

√FORMTEXT +FORMTEXT

¿FORMTEXT ×(√FORMTEXT +FORMTEXT )

√FORMTEXT 2−FORMTEXT 2


¿ FORMTEXT +FORMTEXT √FORMTEXT−FORMTEXT

¿−FORMTEXT−√FORMTEXT


5. 7

√7−3= 7

√7−3× √FORMTEXT+FORMTEXT

√FORMTEXT+FORMTEXT

¿FORMTEXT ×(√FORMTEXT +FORMTEXT )



¿ FORMTEXT +FORMTEXT √FORMTEXT−FORMTEXT


6. √5

√7−3= √5

√7−3× √FORMTEXT +FORMTEXT

√FORMTEXT +FORMTEXT

¿ √FORMTEXT ×(√FORMTEXT+FORMTEXT )


¿ √FORMTEXT+FORMTEXT √FORMTEXTFORMTEXT−FORMTEXT

¿ √FORMTEXT+FORMTEXT √FORMTEXT−FORMTEXT


7. 2√5

√7−3= 2√5

√7−3× √FORMTEXT+FORMTEXT

√FORMTEXT+FORMTEXT

¿FORMTEXT √FORMTEXT ×(√FORMTEXT+FORMTEXT )


¿ FORMTEXT √FORMTEXT +FORMTEXT √FORMTEXTFORMTEXT−FORMTEXT

¿ FORMTEXT √FORMTEXT +FORMTEXT √FORMTEXT−FORMTEXT


8. 2√52√7−3

= 2√52√7−3

× FORMTEXT √FORMTEXT+FORMTEXTFORMTEXT √FORMTEXT+FORMTEXT

¿FORMTEXT √FORMTEXT ×(FORMTEXT √FORMTEXT+FORMTEXT )

¿¿

¿FORMTEXT √FORMTEXT ×(FORMTEXT √FORMTEXT+FORMTEXT )

FORMTEXT−FORMTEXT

¿ FORMTEXT √FORMTEXT +FORMTEXT √FORMTEXTFORMTEXT


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