Surds

What is a Surd?

A surd is an irrational number that is left in its square root form. It is a number that cannot be simplified to remove the square root (or cube root, etc.).

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📑Examples of surds include $\sqrt{3}$ , $\sqrt{7}$ , and $\sqrt{5}$ .

Why do we need them?

Surds are important in mathematics because they allow us to keep our calculations exact. If we tried to write them as decimals, the digits would go on forever without repeating, so we leave them in surd form to maintain precision.

Key Rules of Surds

There are two essential rules you need to know when working with surds:

Rule 1: Multiplication of Surds

\sqrt{a} \times \sqrt{b} = \sqrt{ab}

Explanation: If you multiply one surd by another, you can multiply the numbers inside the square roots and then take the square root of the result.

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📑Example:

\sqrt{7} \times \sqrt{5} = \sqrt{35}

Steps:

Identify the surds you need to multiply.
Multiply the numbers inside the square roots.
Write the result as a single square root.
Rule 2: Simplifying Surds

\sqrt{a} \times \sqrt{a} = a

Explanation: When you multiply a surd by itself, you remove the square root, leaving you with the original number.

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📑Example:

\sqrt{8} \times \sqrt{8} = \sqrt{64}=8

Steps:

Multiply the surd by itself.
Simplify the square root to get the original number.

1. Simplifying Single Surds

Method to Simplify Surds

Identify Square Factors: Split the number under the square root into a product of at least one square number and another factor. Square numbers are: $1,4,9,16,25,36,49,64,81,100,....$
Apply the Square Root: Use the rule a $\sqrt{a\times b} =\sqrt{a}\times \sqrt{b}$ to simplify your surd by taking the square root of the square number.

Worked Examples

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Example 1: Simplify $\sqrt{50}.$ Step 1: Identify the square factor of $50$ .

We ask, "Which square number is a factor of $50$ ?"
$25$ is a square factor because $50=25×2$ . Step 2: Apply the square root.

\sqrt{50} = \sqrt{25 \times 2}

Using the rule:

\sqrt{50} = \sqrt{25} \times \sqrt{2}

\sqrt{25} = 5, \text{ so } \sqrt{50} = 5\sqrt{2}.

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Example 2: Simplify $\sqrt{45}$ . Step 1: Identify the square factor of $45$ .

We ask, "Which square number is a factor of $45$ ?"
$9$ is a square factor because $45=9×5$ . Step 2: Apply the square root.

\sqrt{45} = \sqrt{9 \times 5}

Using the rule:

\sqrt{45} = \sqrt{9} \times \sqrt{5}

\sqrt{9} = 3, \text{ so } \sqrt{45} = 3\sqrt{5}.

2. Simplifying Multiple Surds

When you are asked to simplify an expression that contains more than one surd, you can use the following method to break it down into easier steps.

Method:

Deal with each surd individually: Treat each square root as a separate entity to simplify.
Split the numbers under the square root into a product of at least one square number: Look for factors of the numbers that are perfect squares (like $4, 9, 16, 25$ , etc.).
Use Rule 1 to simplify your answers: Recall that $\sqrt{a}× \sqrt{b}=\sqrt{ab}$ .
When simplifying the whole answer, treat your whole numbers and surds separately: Simplify the coefficients (whole numbers) and the surds on their own, and then combine them.

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Example: Simplify $\sqrt{90} \times \sqrt{20}$

Start by simplifying each surd individually: $90 = 9 \times 10 \quad \text{so} \quad \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}$

$20 = 4 \times 5 \quad \text{so} \quad \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$

Now, multiply the simplified surds together: $3\sqrt{10} \times 2\sqrt{5} = 3 \times 2 \times \sqrt{10} \times \sqrt{5} = 6\sqrt{50}$
Simplify further: $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$

$\text{So, } 6\sqrt{50} = 6 \times 5\sqrt{2} = 30\sqrt{2}$

Thus, the simplified expression for $\sqrt90×\sqrt20\ is\ :success[30\sqrt2]$ .

3. Simplifying Multiple Surds through Division

When dealing with multiple surds that involve division, you can apply the same principles as multiplication but with an additional step for division.

Method:

Simplify each surd individually: Break down each number under the square root into its factors, preferably one of which is a square number.
Apply Rule 1 to simplify your answers: Recall that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ and $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ .
Multiply or divide the surds as appropriate: Simplify the numerator and the denominator separately.
Simplify the entire expression: After multiplying or dividing, simplify the resulting expression further if possible.

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Example: Simplify $\frac{\sqrt{60} \times \sqrt{20}}{\sqrt{12}}$

Start by simplifying each surd individually:

60 = 4 \times 15 \quad \Rightarrow \quad \sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \times \sqrt{15} = 2\sqrt{15}

20 = 4 \times 5 \quad \Rightarrow \quad \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}

12 = 4 \times 3 \quad \Rightarrow \quad \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}

Now, multiply the simplified surds in the numerator:

\sqrt{60} \times \sqrt{20} = 2\sqrt{15} \times 2\sqrt{5} = 4\sqrt{75}

Simplify $\sqrt{75}$

\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}

So, \ 4\sqrt{75} = 4 \times 5\sqrt{3} = 20\sqrt{3}

Finally, divide by the denominator:

\frac{20\sqrt{3}}{2\sqrt{3}} = \frac{20}{2} \times \frac{\sqrt{3}}{\sqrt{3}} = 10 \times 1 = 10

Thus, the simplified expression for $\frac{\sqrt{60} \times \sqrt{20}}{\sqrt{12}}$ is $10$ .

4. Simplifying Surds through Addition and Subtraction

When adding or subtracting surds, you must remember an important rule: You can only add or subtract surds of the same type. This is similar to how you can only add or subtract fractions with the same denominator. If the surds are not of the same type, you will need to simplify them first to see if they can be made the same.

Method:

Simplify the surds: Break down the numbers under the square roots into factors, focusing on square numbers, to simplify them as much as possible.
Check if the surds are of the same type: After simplifying, ensure that the surds are of the same type (i.e., have the same number under the square root).
Add or subtract the surds: Once the surds are of the same type, you can add or subtract them by combining their coefficients (the numbers in front of the surds).

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Example 1: Simplify $\sqrt12+\sqrt27$ Start by simplifying each surd:

12 = 4 \times 3 \quad \text{so} \quad \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}

$27 = 9 \times 3 \quad \text{so} \quad \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$

Now, add the surds:

2\sqrt{3} + 3\sqrt{3} = (2 + 3)\sqrt{3} = 5\sqrt{3}

Thus, the simplified form of $\sqrt12+\sqrt27 \ is \ :success[5\sqrt3]$ .

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Example 2: Simplify $\sqrt63−\sqrt28$ Simplify each surd:

63 = 9 \times 7 \quad \text{so} \quad \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}

28 = 4 \times 7 \quad \text{so} \quad \sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}

Now, subtract the surds:

3\sqrt{7} - 2\sqrt{7} = (3 - 2)\sqrt{7} = 1\sqrt{7} = \sqrt{7}

Thus, the simplified form of $\sqrt63−\sqrt28$ is $\sqrt7$ .

5. What is Rationalising the Denominator?

Rationalising the denominator is a method used in mathematics to eliminate a surd (an irrational number in square-root form) from the bottom of a fraction. Mathematicians consider it tidier and more correct to have no surds in the denominator of a fraction. Therefore, if a surd is present, we "rationalise the denominator" to remove it.

Method:

To rationalise the denominator, you multiply both the top and the bottom of the fraction by the same carefully chosen expression that will eliminate the surd from the bottom.

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Example 1: Rationalising a Single Surd Rationalise the denominator of:

$\frac{2}{\sqrt{3}}$

Step-by-Step Solution:

Identify the surd in the denominator: In this case, $\sqrt3$ is in the denominator.

Multiply both the numerator and the denominator by the surd in the denominator:

\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}

This step is crucial because multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$ is effectively multiplying by $1$ , so the value of the fraction remains the same.

Perform the multiplication:

Numerator:

2 \times \sqrt{3} = 2\sqrt{3}

Denominator:

\sqrt{3} \times \sqrt{3} = \sqrt{9} = 3

So the expression simplifies to:

\frac{2\sqrt{3}}{3}

Final Answer:

\frac{2\sqrt{3}}{3}

The fraction $\frac{2}{\sqrt3}$ has been rationalised to $\frac{2\sqrt{3}}{3}$ , which has no surd in the denominator.

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Example 2: Rationalising the Denominator with Other Numbers Rationalise the denominator of:

$\frac{5}{3 - \sqrt{2}}$

Trick:

For questions like this, the trick is to multiply both the top and the bottom of the fraction by the same expression as the bottom, but with the sign changed. This technique uses the difference of two squares.

Step-by-Step Solution:

Identify the denominator and its conjugate:

The denominator is $3−\sqrt2$ .
The conjugate of the denominator is $3+\sqrt2$ . Multiply both the numerator and the denominator by the conjugate:

\frac{5}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}}

Multiply the numerator (Tops):

5 \times (3 + \sqrt{2}) = 15 + 5\sqrt{2}

Multiply the denominator (Bottoms) using FOIL (First, Outer, Inner, Last):

First: $3 \times 3 = 9$
Outer: $3 \times \sqrt{2} = 3\sqrt{2}$
Inner: $\sqrt{2} \times 3 = -3\sqrt{2}$
Last: $-\sqrt{2} \times \sqrt{2} = -2$

So the denominator becomes:

9 + 3\sqrt{2} - 3\sqrt{2} - 2

Simplify the denominator:

The middle terms $3\sqrt2$ and $−3\sqrt2$ cancel each other out.
The simplified denominator is:

9 - 2 = 7

Final Answer: The rationalised form of the expression is:

\frac{15 + 5\sqrt{2}}{7}

Surds Simplified Revision Notes for GCSE OCR Maths

Surds

What is a Surd?

Why do we need them?

Key Rules of Surds

1. Simplifying Single Surds

Method to Simplify Surds

Worked Examples

2. Simplifying Multiple Surds

3. Simplifying Multiple Surds through Division

4. Simplifying Surds through Addition and Subtraction

5. What is Rationalising the Denominator?

Method:

500K+ Students Use These Powerful Tools to Master Surds For their GCSE Exams.

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