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Addition and Subtraction of Surds

In addition and subtraction of surds we will learn how to find the sum or difference of two or more surds only when they are in the simplest form of like surds.

For addition and subtraction of surds, we have to check the surds that if they are similar surds or dissimilar surds.

Follow the following steps to find the addition and subtraction of two or more surds:

Step I: Convert each surd in its simplest mixed form.

Step II: Then find the sum or difference of rational co-efficient of like surds.

Step III: Finally, to get the required sum or difference of like surds multiply the result obtained in step II by the surd-factor of like surds.

Step IV: The sum or difference of unlike surds is expressed in a number of terms by connecting them with positive sign (+) or negative (-) sign.

If the surds are similar, then we can sum or subtract rational coefficients to find out the result of addition or subtraction.

anx±bnx=(a±b)nx

The above equation shows the rule of addition and subtraction of surds where irrational factor is nx and a, b are rational coefficients.

Surds firstly need to be expressed in their simplest form or lowest order with minimum radicand, and then only we can find out which surds are similar. If the surds are similar, we can add or subtract them according to the rule mentioned above.

For example we need to find the addition of 28, 218.

Both surds are in same order. Now we need find express them in their simplest form.

So 28 = 24×2 = 222×2 = 222

And 218 = 29×2 = 232×2 = 322.

As both surds are similar, we can add their rational co-efficient and find the result. 

Now 28 + 218 = 222 + 322 = 522.

Similarly we will find out subtraction of 275, 248.

275= 225×3= 252×3= 523

248 = 216×3 = 242×3= 423

So 275 - 248 = 523 - 423 = 23.

But if we need to find out the addition or subtraction of 322 and 223, we can only write it as 322 + 223 or 322 - 223. As the surds are dissimilar, further addition and subtraction are not possible in surd forms.

Examples of Addition and Subtraction of Surds:

1. Find the sum of √12 and √27.

Solution:

Sum of √12 and √27

= √12 + √27

Step I: Express each surd in its simplest mixed form;

= 223 + 333

= 2√3 + 3√3

Step II: Then find the sum of rational co-efficient of like surds.

= 5√3


2. Simplify 3232 + 6245 - 2162 - 22245.

Solution:

3232 + 6245 - 2162 - 22245

= 3216×2 + 629×5 - 281×2 - 2249×5

= 3242×2 + 6232×5 - 292×2 - 2272×5

= 1222 + 1825 - 922 - 1425

= 322 + 425


3. Subtract 2√45 from 4√20.

Solution:

Subtract 2√45 from 4√20

= 4√20 - 2√45

Now convert each surd in its simplest form

= 4225 - 2335

= 8√5 - 6√5

Clearly, we see that 8√5 and 6√5 are like surds.

Now find the difference of rational co-efficient of like surds

= 2√5.


4. Simplify 73128 + 53375 - 2354 - 231029.

Solution:

73128 + 53375 - 2354 - 231029

= 7364×2 + 53125×3 - 327×2 - 23343×3

= 7343×2 + 5353×3 - 333×2 - 2373×3

= 2832 + 2533 - 332 - 1433

= 2532 + 1133.


5. Simplify: 5√8 - √2 + 5√50 - 25/2

Solution:

5√8 - √2 + 5√50 - 25/2

Now convert each surd in its simplest form

= 5222 - √2 + 5255 - 25

= 5222 - √2 + 5255 - 22222

= 10√2 - √2 + 25√2 - 4√2

Clearly, we see that 8√5 and 6√5 are like surds.

Now find the sum and difference of rational co-efficient of like surds

= 30√2


6. Simplify 2433 + 5324 - 2228 - 4263.

Solution:

2433 + 5324 - 2228 - 4263

= 2433 + 538×3 - 224×7 - 429×7

2433 + 5323×3 - 2222×7 - 4232×7

= 2433 + 1033 - 427 - 1227

= 3433 - 1627.


7. Simplify: 2∛5 - ∛54 + 3∛16 - ∛625

Solution:

2∛5 - ∛54 + 3∛16 - ∛625

Now convert each surd in its simplest form

= 2∛5 - 32333 + 332222 - 35555

= 2∛5 - 3∛2 + 6∛2 - 5∛5

= (6∛2 - 3∛2) + (2∛5 - 5∛5), [Combining the like surds]

Now find the difference of rational co-efficient of like surds

= 3∛2 - 3∛5


8. Simplify 527 + 3220 - 2280 - 3284.

Solution:

527 + 3220 - 2280 - 3284

= 527 + 324×5 - 2216×5 - 3216×6

= 527 + 3222×5 - 2242×2 - 3242×6

= 527 + 625 - 825 - 1226

= 527 - 225 - 1226.


Note:

√x + √y ≠ x+y and

√x - √y ≠ xy

 Surds






11 and 12 Grade Math

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