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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.
anβxΒ±bnβx=(aΒ±b)nβx
The above equation shows the rule of addition and subtraction of surds where irrational factor is nβx 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 2β8, 2β18.
Both surds are in same order. Now we need find express them in their simplest form.
So 2β8 = 2β4Γ2 = 2β22Γ2 = 22β2
And 2β18 = 2β9Γ2 = 2β32Γ2 = 32β2.
As both surds are similar, we can add their rational co-efficient and find the result.
Now 2β8 + 2β18 = 22β2 + 32β2 = 52β2.
Similarly we will find out subtraction of 2β75, 2β48.
2β75= 2β25Γ3= 2β52Γ3= 52β3
2β48 = 2β16Γ3 = 2β42Γ3= 42β3
So 2β75 - 2β48 = 52β3 - 42β3 = 2β3.
But if we need to find out the addition or subtraction of 32β2 and 22β3, we can only write it as 32β2 + 22β3 or 32β2 - 22β3. 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 β12 and β27.
Solution:
Sum of β12 and β27
= β12 + β27
Step I: Express each surd in its simplest mixed form;
= β2β 2β 3 + β3β 3β 3
= 2β3 + 3β3
Step II: Then find the sum of rational co-efficient of like surds.
= 5β3
2. Simplify 32β32 + 62β45 - 2β162 - 22β245.
Solution:
32β32 + 62β45 - 2β162 - 22β245
= 32β16Γ2 + 62β9Γ5 - 2β81Γ2 - 22β49Γ5
= 32β42Γ2 + 62β32Γ5 - 2β92Γ2 - 22β72Γ5
= 122β2 + 182β5 - 92β2 - 142β5
= 32β2 + 42β5
3. Subtract 2β45 from 4β20.
Solution:
Subtract 2β45 from 4β20
= 4β20 - 2β45
Now convert each surd in its simplest form
= 4β2β 2β 5 - 2β3β 3β 5
= 8β5 - 6β5
Clearly, we see that 8β5 and 6β5 are like surds.
Now find the difference of rational co-efficient of like surds
= 2β5.
4. Simplify 73β128 + 53β375 - 23β54 - 23β1029.
Solution:
73β128 + 53β375 - 23β54 - 23β1029
= 73β64Γ2 + 53β125Γ3 - 3β27Γ2 - 23β343Γ3
= 73β43Γ2 + 53β53Γ3 - 3β33Γ2 - 23β73Γ3
= 283β2 + 253β3 - 33β2 - 143β3
= 253β2 + 113β3.
5. Simplify: 5β8 - β2 + 5β50 - 25/2
Solution:
5β8 - β2 + 5β50 - 25/2
Now convert each surd in its simplest form
= 5β2β 2β 2 - β2 + 5β2β 5β 5 - β25
= 5β2β 2β 2 - β2 + 5β2β 5β 5 - β2β 2β 2β 2β 2
= 10β2 - β2 + 25β2 - 4β2
Clearly, we see that 8β5 and 6β5 are like surds.
Now find the sum and difference of rational co-efficient of like surds
= 30β2
6. Simplify 243β3 + 53β24 - 22β28 - 42β63.
Solution:
243β3 + 53β24 - 22β28 - 42β63
= 243β3 + 53β8Γ3 - 22β4Γ7 - 42β9Γ7
= 243β3 + 53β23Γ3 - 22β22Γ7 - 42β32Γ7
= 243β3 + 103β3 - 42β7 - 122β7
= 343β3 - 162β7.
7. Simplify: 2β5 - β54 + 3β16 - β625
Solution:
2β5 - β54 + 3β16 - β625
Now convert each surd in its simplest form
= 2β5 - 3β2β 3β 3β 3 + 33β2β 2β 2β 2 - 3β5β 5β 5β 5
= 2β5 - 3β2 + 6β2 - 5β5
= (6β2 - 3β2) + (2β5 - 5β5), [Combining the like surds]
Now find the difference of rational co-efficient of like surds
= 3β2 - 3β5
8. Simplify 52β7 + 32β20 - 22β80 - 32β84.
Solution:
52β7 + 32β20 - 22β80 - 32β84
= 52β7 + 32β4Γ5 - 22β16Γ5 - 32β16Γ6
= 52β7 + 32β22Γ5 - 22β42Γ2 - 32β42Γ6
= 52β7 + 62β5 - 82β5 - 122β6
= 52β7 - 22β5 - 122β6.
Note:
βx + βy β βx+y and
βx - βy β βxβy
β Surds
11 and 12 Grade Math
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