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Indices and surds

Surds

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Summary

Surds

In a nutshell

A surd is the square root of a number that results in an irrational number. Writing a number in surd form keeps the values accurate for a calculation, as the irrational numbers then do not need to be rounded to the nearest decimal. There are different rules for manipulating expressions with surds, including how to simplify, add and subtract them.



Definition

A surd is a root that cannot be simplified into a whole or rational number. They are irrational numbers, which cannot be written as a fraction and their decimal form goes on forever.


Example 1

Is 2\sqrt22​ a surd?


2\sqrt22​ is an irrational number.​

2=1.41421356237...\sqrt2=1.41421356237...2​=1.41421356237... 


Therefore, yes, 2\sqrt22​ is a surd.


Example 2

Is 9\sqrt99​ a surd?


9=3\sqrt9 = 39​=3 is an integer.


Therefore, no, 9 \sqrt{9}9​ is not a surd.

​


The rules of surds

These are the rules for surds which are derived from the laws of indices:


a×b=aba÷b=aba×a=a\boxed{\begin{aligned}\sqrt{a}\times \sqrt{b} = \sqrt{ab}\\\\\sqrt{a}\div \sqrt{b} = \sqrt{\frac{a}b}\\\\\sqrt{a}\times \sqrt{a} = a\end{aligned}}a​×b​=ab​a​÷b​=ba​​a​×a​=a​​​​


Example 3

Simplify 2×8\sqrt{2}\times \sqrt82​×8​.

​2×8=16=4‾\begin{aligned}\sqrt{2}\times \sqrt8&= \sqrt{16} \\&=\underline{4}\end{aligned} 2​×8​​=16​=4​​​​


Example 4

Simplify 200÷8\sqrt{200} \div \sqrt8200​÷8​.

200÷8=25=5‾\begin{aligned}\sqrt{200} \div \sqrt8&= \sqrt{25} \\&=\underline{5}\end{aligned} 200​÷8​​=25​=5​​​​



Simplifying surds

A surd is in its simplest form when the number within the root has no square numbers as factors. So when simplifying a surd, you are extracting the square numbers from underneath the root, leaving the smallest possible number.


Procedure

1.

Find square numbers that are factors of the number within the root.

2.

Rewrite the surd as the product of the square number(s) and the other number(s).

3.

Root the square number(s) and write it outside of the root, leaving the other number(s) inside.


Example 5

Simplify 20\sqrt{20}20​.


444 is a square number and a factor of 202020.

20=4×5=25‾\begin{aligned}\sqrt{20}&= \sqrt{4\times 5} \\&=\underline{2\sqrt5}\end{aligned} 20​​=4×5​=25​​​​​


Example 6

Simplify 300+27\sqrt{300}+\sqrt{27}300​+27​.


100100100 is a square factor of 300300300 and 999 is a square factor of 333.​

300+27=100×3+9×3=103+33=133‾\begin{aligned}\sqrt{300}+\sqrt{27}&= \sqrt{100\times3}+\sqrt{9\times3} \\&=10\sqrt3+3\sqrt3\\&=\underline{13\sqrt3}\end{aligned} 300​+27​​=100×3​+9×3​=103​+33​=133​​​


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Surds: Simplify, add and subtract - Higher

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FAQs - Frequently Asked Questions

Is the square root of 9 a surd?

The square root of 9 is 3, which is an integer. So, that means the square root of 9 is not a surd.

What is an example of a surd?

The square root of 2 is an irrational number and is therefore a surd.

What is a surd?

A surd is a square root that cannot be simplified into a whole or rational number.

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