Surds

Introduction

• Surds are numbers written in square root form that produce irrational numbers.
• They are used when we work with roots in their exact form.
• In many maths problems, we need to keep numbers in root form instead of changing them into decimals.

Example:

$\sqrt{5}$ is a surd because it is an irrational number and cannot be written as a simple fraction.

What are Rational and Irrational Numbers?

In mathematics, numbers are divided into rational and irrational numbers.

Rational numbers:

Numbers that can be written as a simple fraction $\frac{a}{b}$ (where $b\neq0$) or as a whole number are called rational numbers.

Example:

• $\frac{3}{4}=0.75$
• $0.333…=\frac{1}{3}$
• $100$, $0$ etc.
• $-3$, $-2$ etc.

Irrational numbers:

Numbers that cannot be written as a simple fraction $\frac{a}{b}$ (where $b\neq0$) are called irrational numbers.

Example:

• $\sqrt{5}$
• $\sqrt{2}\approx1.4142…$
• $\pi\approx3.1415…$
• $\sqrt{3}\approx1.7320…$

What are Surds?

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

• Surds usually occur when we take the root of numbers that are not perfect powers.
• These roots can be square roots, cube roots, or other roots. Instead of writing them as long decimal numbers, we keep them in root form to show the exact value.

Example:

• $\sqrt[3]{2} \longrightarrow$ This is also a surd because it cannot be written as a whole number.
• $\sqrt{5} \longrightarrow$ This is a surd because $5$ is not a perfect square.
• $\sqrt{16}=4 \longrightarrow$ This is not a surd because $16$ is a perfect square and the root gives a whole number.
• $\sqrt{9}=3 \longrightarrow$ This is not a surd because $9$ is a perfect square and the root gives a whole number.

How to simplify Surds?

Simplifying surds means taking out perfect square factors from inside the square root.

Steps to Simplify Surds:

Solved Example

Simplify $\sqrt{288}$

SOLUTION

1

Break the number into factors using a factor tree.

So,

$$288=2\times2\times2\times2\times2\times3\times3$$

2

Group the same numbers into pairs.

$$(2\times2)(2\times2)(3\times3)\times2$$

3

Use root and take pairs outside

$$\sqrt{288}=\sqrt{(2\times2)(2\times2)(3\times3)\times2}$$

$$\sqrt{288}=2\times2\times3\sqrt{2}$$

$$\sqrt{288}=12\sqrt{2}$$

Final Answer: $12\sqrt{2}$

Solved Example

Simplify $5\sqrt{125}$

SOLUTION

1

Break the number into factors using a factor tree.

So,

$$125=5\times5\times5$$

2

Group into pairs

$$(5\times5)\times5$$

3

Use root and take pairs outside

$$5\sqrt{125}=5\sqrt{(5\times5)\times5}$$

$$5\sqrt{125}=5\times5\sqrt{5}$$

$$5\sqrt{125}=25\sqrt{5}$$

Final Answer: $25\sqrt{5}$

How to Add and Subtract Surds?

Surds can be added or subtracted only when they have the same root.

Steps to Add and Subtract Surds:

How to Multiply and Divide Surds?

Surds can be multiplied or divided by multiplying or dividing the numbers outside the roots and the numbers inside the roots, while keeping the answer in root form.

Steps to Multiply and Divide Surds:

How to rationalise the denominator?

Rationalising the denominator means removing the surd from the denominator of a fraction so that the denominator becomes a rational number.

Steps to rationalise the denominator:

Solved Examples

Solved Example

Problem 1: $\sqrt{200}-\sqrt{72}$

SOLUTION

1

Simplify the surds

• For $\sqrt{200};$

$$\sqrt{200}=\sqrt{(10\times10)\times2}=10\sqrt{2}$$

• For $\sqrt{72}:$

$$\sqrt{72}=\sqrt{(6\times6)\times2}=6\sqrt{2}$$

2

Subtract the coefficients

$$10\sqrt{2}-6\sqrt{2}=(10-6)\sqrt{2}$$

3

Keep the root same

$$=4\sqrt{2}$$

Final Answer: $4\sqrt{2}$

Solved Example

Problem 2: Simplify $\sqrt{80}-\frac{10}{\sqrt{5}}$

SOLUTION

1

Simplify the surds

$$\sqrt{80}=\sqrt{16\times5}=4\sqrt{5}$$

2

Rationalise the denominator

$$\frac{10}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}}=\frac{10\sqrt{5}}{5}$$

$$=2\sqrt{5}$$

Final Answer: $2\sqrt{5}$

Solved Example

Problem 3: $\sqrt{300}-2\sqrt{x}=4\sqrt{3}$

SOLUTION

1

Break the number into factors using a factor tree.

$$\sqrt{300}=\sqrt{100\times3}=10\sqrt{3}$$

2

Substitute and simplify equation

$$10\sqrt{3}-2\sqrt{x}=4\sqrt{3}$$

3

Rearrange and solve

$$10\sqrt{3}-4\sqrt{3}=2\sqrt{x}$$

$$6\sqrt{3}=2\sqrt{x}$$

Divide both sides by $2$:

$$3\sqrt{3}=\sqrt{x}$$

4

Square both sides

$$(3\sqrt{3})^{2}=x$$

$$=9\times3=27$$

Final Answer: $x = 27$