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.

\(a\sqrt[n]{x}\pm b\sqrt[n]{x} = (a\pm b)\sqrt[n]{x}\)

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 co-efficient 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 co-efficient 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 co-efficient 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 co-efficient 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 co-efficient 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

From Addition and Subtraction of Surds to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. 3-digit Numbers on an Abacus | Learning Three Digit Numbers | Math

    Oct 08, 24 10:53 AM

    3-Digit Numbers on an Abacus
    We already know about hundreds, tens and ones. Now let us learn how to represent 3-digit numbers on an abacus. We know, an abacus is a tool or a toy for counting. An abacus which has three rods.

    Read More

  2. Names of Three Digit Numbers | Place Value |2- Digit Numbers|Worksheet

    Oct 07, 24 04:07 PM

    How to write the names of three digit numbers? (i) The name of one-digit numbers are according to the names of the digits 1 (one), 2 (two), 3 (three), 4 (four), 5 (five), 6 (six), 7 (seven)

    Read More

  3. Worksheets on Number Names | Printable Math Worksheets for Kids

    Oct 07, 24 03:29 PM

    Traceable math worksheets on number names for kids in words from one to ten will be very helpful so that kids can practice the easy way to read each numbers in words.

    Read More

  4. The Number 100 | One Hundred | The Smallest 3 Digit Number | Math

    Oct 07, 24 03:13 PM

    The Number 100
    The greatest 1-digit number is 9 The greatest 2-digit number is 99 The smallest 1-digit number is 0 The smallest 2-digit number is 10 If we add 1 to the greatest number, we get the smallest number of…

    Read More

  5. Missing Numbers Worksheet | Missing Numerals |Free Worksheets for Kids

    Oct 07, 24 12:01 PM

    Missing numbers
    Math practice on missing numbers worksheet will help the kids to know the numbers serially. Kids find difficult to memorize the numbers from 1 to 100 in the age of primary, we can understand the menta

    Read More