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 coefficient 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 surdfactor 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.
\(a\sqrt[n]{x}\pm b\sqrt[n]{x} = (a\pm b)\sqrt[n]{x}\)
The above equation shows the rule of addition and subtraction of surds where irrational factor is \(\sqrt[n]{x}\) 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 \(\sqrt[2]{8}\), \(\sqrt[2]{18}\).
Both surds are in same order. Now we need find express them in their simplest form.
So \(\sqrt[2]{8}\) = \(\sqrt[2]{4\times 2}\) = \(\sqrt[2]{2^{2}\times 2}\) = \(2\sqrt[2]{2}\)
And \(\sqrt[2]{18}\) = \(\sqrt[2]{9\times 2}\) = \(\sqrt[2]{3^{2}\times 2}\) = \(3\sqrt[2]{2}\).
As both surds are similar, we can add their rational coefficient and find the result.
Now \(\sqrt[2]{8}\) + \(\sqrt[2]{18}\) = \(2\sqrt[2]{2}\) + \(3\sqrt[2]{2}\) = \(5\sqrt[2]{2}\).
Similarly we will find out subtraction of \(\sqrt[2]{75}\), \(\sqrt[2]{48}\).
\(\sqrt[2]{75}\)= \(\sqrt[2]{25\times 3}\)= \(\sqrt[2]{5^{2}\times 3}\)= \(5\sqrt[2]{3}\)
\(\sqrt[2]{48}\) = \(\sqrt[2]{16\times 3}\) = \(\sqrt[2]{4^{2}\times 3}\)= \(4\sqrt[2]{3}\)
So \(\sqrt[2]{75}\)  \(\sqrt[2]{48}\) = \(5\sqrt[2]{3}\)  \(4\sqrt[2]{3}\) = \(\sqrt[2]{3}\).
But if we need to find out the addition or subtraction of \(3\sqrt[2]{2}\) and \(2\sqrt[2]{3}\), we can only write it as \(3\sqrt[2]{2}\) + \(2\sqrt[2]{3}\) or \(3\sqrt[2]{2}\)  \(2\sqrt[2]{3}\). 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;
= \(\sqrt{2\cdot 2\cdot 3}\) + \(\sqrt{3\cdot 3\cdot 3}\)
= 2√3 + 3√3
Step II: Then find the sum of rational coefficient of like surds.
= 5√3
2. Simplify \(3\sqrt[2]{32}\) + \(6\sqrt[2]{45}\)  \(\sqrt[2]{162}\)  \(2\sqrt[2]{245}\).
Solution:
\(3\sqrt[2]{32}\) + \(6\sqrt[2]{45}\)  \(\sqrt[2]{162}\)  \(2\sqrt[2]{245}\)
= \(3\sqrt[2]{16\times 2}\) + \(6\sqrt[2]{9\times 5}\)  \(\sqrt[2]{81\times 2}\)  \(2\sqrt[2]{49\times 5}\)
= \(3\sqrt[2]{4^{2}\times 2}\) + \(6\sqrt[2]{3^{2}\times 5}\)  \(\sqrt[2]{9^{2}\times 2}\)  \(2\sqrt[2]{7^{2}\times 5}\)
= \(12\sqrt[2]{2}\) + \(18\sqrt[2]{5}\)  \(9\sqrt[2]{2}\)  \(14\sqrt[2]{5}\)
= \(3\sqrt[2]{2}\) + \(4\sqrt[2]{5}\)
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
= 4\(\sqrt{2\cdot 2\cdot 5}\)  2\(\sqrt{3\cdot 3\cdot 5}\)
= 8√5  6√5
Clearly, we see that 8√5 and 6√5 are like surds.
Now find the difference of rational coefficient of like surds
= 2√5.
4. Simplify \(7\sqrt[3]{128}\) + \(5\sqrt[3]{375}\)  \(2\sqrt[3]{54}\)  \(2\sqrt[3]{1029}\).
Solution:
\(7\sqrt[3]{128}\) + \(5\sqrt[3]{375}\)  \(2\sqrt[3]{54}\)  \(2\sqrt[3]{1029}\)
= \(7\sqrt[3]{64\times 2}\) + \(5\sqrt[3]{125\times 3}\)  \(\sqrt[3]{27\times 2}\)  \(2\sqrt[3]{343\times 3}\)
= \(7\sqrt[3]{4^{3}\times 2}\) + \(5\sqrt[3]{5^{3}\times 3}\)  \(\sqrt[3]{3^{3}\times 2}\)  \(2\sqrt[3]{7^{3}\times 3}\)
= \(28\sqrt[3]{2}\) + \(25\sqrt[3]{3}\)  \(3\sqrt[3]{2}\)  \(14\sqrt[3]{3}\)
= \(25\sqrt[3]{2}\) + \(11\sqrt[3]{3}\).
5. Simplify: 5√8  √2 + 5√50  2\(^{5/2}\)
Solution:
5√8  √2 + 5√50  2\(^{5/2}\)
Now convert each surd in its simplest form
= 5\(\sqrt{2\cdot 2\cdot 2}\)  √2 + 5\(\sqrt{2\cdot 5\cdot 5}\)  \(\sqrt{2^{5}}\)
= 5\(\sqrt{2\cdot 2\cdot 2}\)  √2 + 5\(\sqrt{2\cdot 5\cdot 5}\)  \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 2}\)
= 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 coefficient of like surds
= 30√2
6. Simplify \(24\sqrt[3]{3}\) + \(5\sqrt[3]{24}\)  \(2\sqrt[2]{28}\)  \(4\sqrt[2]{63}\).
Solution:
\(24\sqrt[3]{3}\) + \(5\sqrt[3]{24}\)  \(2\sqrt[2]{28}\)  \(4\sqrt[2]{63}\)
= \(24\sqrt[3]{3}\) + \(5\sqrt[3]{8\times 3}\)  \(2\sqrt[2]{4\times 7}\)  \(4\sqrt[2]{9\times 7}\)
= \(24\sqrt[3]{3}\) + \(5\sqrt[3]{2^{3}\times 3}\)  \(2\sqrt[2]{2^{2}\times 7}\)  \(4\sqrt[2]{3^{2}\times 7}\)
= \(24\sqrt[3]{3}\) + \(10\sqrt[3]{3}\)  \(4\sqrt[2]{7}\)  \(12\sqrt[2]{7}\)
= \(34\sqrt[3]{3}\)  \(16\sqrt[2]{7}\).
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  \(\sqrt[3]{2\cdot 3\cdot 3\cdot 3}\) + 3\(\sqrt[3]{2\cdot 2\cdot 2\cdot 2}\)  \(\sqrt[3]{5\cdot 5\cdot 5\cdot 5}\)
= 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 coefficient of like surds
= 3∛2  3∛5
8. Simplify \(5\sqrt[2]{7}\) + \(3\sqrt[2]{20}\)  \(2\sqrt[2]{80}\)  \(3\sqrt[2]{84}\).
Solution:
\(5\sqrt[2]{7}\) + \(3\sqrt[2]{20}\)  \(2\sqrt[2]{80}\)  \(3\sqrt[2]{84}\)
= \(5\sqrt[2]{7}\) + \(3\sqrt[2]{4\times 5}\)  \(2\sqrt[2]{16\times 5}\)  \(3\sqrt[2]{16\times 6}\)
= \(5\sqrt[2]{7}\) + \(3\sqrt[2]{2^{2}\times 5}\)  \(2\sqrt[2]{4^{2}\times 2}\)  \(3\sqrt[2]{4^{2}\times 6}\)
= \(5\sqrt[2]{7}\) + \(6\sqrt[2]{5}\)  \(8\sqrt[2]{5}\)  \(12\sqrt[2]{6}\)
= \(5\sqrt[2]{7}\)  \(2\sqrt[2]{5}\)  \(12\sqrt[2]{6}\).
Note:
√x + √y ≠ \(\sqrt{x + y}\) and
√x  √y ≠ \(\sqrt{x  y}\)
`● Surds
11 and 12 Grade Math
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