# 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.

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.

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. 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.

3. 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

4. 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

Note:

√x + √y ≠ $$\sqrt{x + y}$$ and

√x - √y ≠ $$\sqrt{x - y}$$