the sum of two surds
Answers
Answer:
Step-by-step explanation:
If the surds are similar, then we can sum or subtract rational coefficients to find out the result of addition or subtraction.
ax−−√n±bx−−√n=(a±b)x−−√n
The above equation shows the rule of addition and subtraction of surds where irrational factor is x−−√n 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 8–√2, 18−−√2.
Both surds are in same order. Now we need find express them in their simplest form.
So 8–√2 = 4×2−−−−√2 = 22×2−−−−−√2 = 22–√2
And 18−−√2 = 9×2−−−−√2 = 32×2−−−−−√2 = 32–√2.
As both surds are similar, we can add their rational co-efficient and find the result.
Now 8–√2 + 18−−√2 = 22–√2 + 32–√2 = 52–√2.
Similarly we will find out subtraction of 75−−√2, 48−−√2.
75−−√2= 25×3−−−−−√2= 52×3−−−−−√2= 53–√2
48−−√2 = 16×3−−−−−√2 = 42×3−−−−−√2= 43–√2
So 75−−√2 - 48−−√2 = 53–√2 - 43–√2 = 3–√2.
But if we need to find out the addition or subtraction of 32–√2 and 23–√2, we can only write it as 32–√2 + 23–√2 or 32–√2 - 23–√2. As the surds are dissimilar, further addition and subtraction are not possible in surd forms.
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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;
= 2⋅2⋅3−−−−−−√ + 3⋅3⋅3−−−−−−√
= 2√3 + 3√3
Step II: Then find the sum of rational co-efficient of like surds.
= 5√3
2. Simplify 332−−√2 + 645−−√2 - 162−−−√2 - 2245−−−√2.
Solution:
332−−√2 + 645−−√2 - 162−−−√2 - 2245−−−√2
= 316×2−−−−−√2 + 69×5−−−−√2 - 81×2−−−−−√2 - 249×5−−−−−√2
= 342×2−−−−−√2 + 632×5−−−−−√2 - 92×2−−−−−√2 - 272×5−−−−−√2
= 122–√2 + 185–√2 - 92–√2 - 145–√2
= 32–√2 + 45–√2
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
= 42⋅2⋅5−−−−−−√ - 23⋅3⋅5−−−−−−√
= 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 7128−−−√3 + 5375−−−√3 - 254−−√3 - 21029−−−−√3.
Solution:
7128−−−√3 + 5375−−−√3 - 254−−√3 - 21029−−−−√3
= 764×2−−−−−√3 + 5125×3−−−−−−√3 - 27×2−−−−−√3 - 2343×3−−−−−−√3
= 743×2−−−−−√3 + 553×3−−−−−√3 - 33×2−−−−−√3 - 273×3−−−−−√3
= 282–√3 + 253–√3 - 32–√3 - 143–√3
= 252–√3 + 113–√3.
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
= 52⋅2⋅2−−−−−−√ - √2 + 52⋅5⋅5−−−−−−√ - 25−−√
= 52⋅2⋅2−−−−−−√ - √2 + 52⋅5⋅5−−−−−−√ - 2⋅2⋅2⋅2⋅2−−−−−−−−−−√
= 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 243–√3 + 524−−√3 - 228−−√2 - 463−−√2.
Solution:
243–√3 + 524−−√3 - 228−−√2 - 463−−√2
= 243–√3 + 58×3−−−−√3 - 24×7−−−−√2 - 49×7−−−−√2
= 243–√3 + 523×3−−−−−√3 - 222×7−−−−−√2 - 432×7−−−−−√2
= 243–√3 + 103–√3 - 47–√2 - 127–√2
= 343–√3 - 167–√2.
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 - 2⋅3⋅3⋅3−−−−−−−−√3 + 32⋅2⋅2⋅2−−−−−−−−√3 - 5⋅5⋅5⋅5−−−−−−−−√3
= 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 57–√2 + 320−−√2 - 280−−√2 - 384−−√2.
Solution:
57–√2 + 320−−√2 - 280−−√2 - 384−−√2
= 57–√2 + 34×5−−−−√2 - 216×5−−−−−√2 - 316×6−−−−−√2
= 57–√2 + 322×5−−−−−√2 - 242×2−−−−−√2 - 342×6−−−−−√2
= 57–√2 + 65–√2 - 85–√2 - 126–√2
= 57–√2 - 25–√2 - 126–√2.