01.05 Surds
In this lesson you will learn how to:
- Simplify surds
- Add, subtract and multiply surds
Prerequisites from previous chapters:
- N/A
VIDEO LESSON
INTERACTIVE SELF-STUDY
A surd is any number that can only be expressed using roots.
Examples of surds are \(\sqrt{2}\), \(\sqrt{19}\), \(5\sqrt{2}\), \(2+\sqrt{5}\), \(\sqrt{6}-\sqrt{2}\) and \(\sqrt[3]{5}+\sqrt{10}\)
We will primarily focus on surds that are square roots of integers
Surds are examples of irrational numbers
- Irrational numbers are numbers that cannot be written as the ratio (fraction) of two integers.
In other words, they cannot be written in the form \(\displaystyle \frac{p}{q}\), where \(p\) and \(q\) are integers - (In Year 2 Pure you will learn how to prove that \(\sqrt{2}\) is irrational)
Since surds are irrational numbers, their decimal expansions are never-ending and never repeats.
For example \(\sqrt{2}=1.4142135…\)
LAWS OF SURDS AND SIMPLIFYING SURDS
Since an \(n\)-th root just means raising to a fractional power, surds also follow the laws of indices
Consider \(\sqrt{ab}\) and \(\displaystyle \sqrt{\frac{a}{b}}\)
i. Express \(\sqrt{ab}\) as a power of \(ab\)
iii. Express \(\displaystyle \sqrt{\frac{a}{b}}\) as a power of \(\displaystyle \frac{a}{b}\)
iv. Express your answer to part iii as a power of \(a\) and power of \(b\) and hence express it in terms of \(\sqrt{a}\) and \(\sqrt{b}\)
- \(\sqrt{ab}=\sqrt{a}\sqrt{b}\phantom{–}\) for \(a>0,\phantom{–}\) \(b>0\)
- \(\displaystyle \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\phantom{–}\) for \(a>0,\phantom{–}\) \(b>0\)
We can use the fact \(\sqrt{ab}=\sqrt{a}\sqrt{b}\) to simplify surds.
A surd can be simplified if you can express it in terms of another surd with a smaller number under the root symbol
To determine whether a surd can be simplified, you should aim to write the number under the root sign as the product of a square number and some other number. Then split the surd into the product of two surds
Worked Example 1 – (GRADE E)
Simplify \(\sqrt{72}\)
Step 1: Find a square number that is a factor of the number under the root sign
\(4, 9\) and \(36\) are all perfect squares that are factors of \(72\)
It is best to use the largest square number to simplify the surd otherwise you will need to perform multiple steps.
\(72=36\times 2\)
Step 2: Express the surd as a product of surds
\(\sqrt{72}=\sqrt{36\times 2}=\sqrt{36}\sqrt{2}\)
Step 3: Simplify the surd
\(\sqrt{36}\sqrt{2}=6\sqrt{2}\)
Example 2 – (GRADE E)
Simplify the following surds
a. \(\sqrt{75}\)
c. \(\sqrt{98}\)
d. \(\displaystyle \frac{\sqrt{56}}{\sqrt{14}}\)
We can also go in the reverse direction and express surds of the form \(m\sqrt{a}\) in the form \(\sqrt{n}\)
We can write \(m\) as \(\sqrt{m^2}\). Then \(m\sqrt{a}=\sqrt{m^2}\sqrt{a}=\sqrt{m^2a}\)
Example 3 – (GRADE E)
a. Express the following surds in the form \(\sqrt{n}\)
\(\phantom{-}\) i. \(4\sqrt{2}\) \(\phantom{-}\) ii. \(3\sqrt{3}\) \(\phantom{-}\) iii. \(2 \sqrt{7}\)
ADDING, SUBTRACTING AND MULTIPLYING SURDS
\(\sqrt{a}\) represents a single value, so it can be treated like any algebraic expression.
- \(m\sqrt{a}±n\sqrt{a}=(m±n)\sqrt{a}\)
(Similar to how \(mx±nx=(m±n)x\))
- \(\sqrt{a} \times \sqrt{a} = (\sqrt{a})^2=a\)
(Similar to how \(x \times x=x^2\))
We can use these facts to help us manipulate surds
Worked Example 4 – (GRADE E)
Simplify \(\sqrt{180}-\sqrt{80}\)
\(\sqrt{180}=\sqrt{36 \times 5}=\sqrt{36}\sqrt{5}=6\sqrt{5}\)
\(\sqrt{80}=\sqrt{16\times 5}=\sqrt{16}\sqrt{5}=4\sqrt{5}\)
Hence \(\sqrt{180}-\sqrt{80}=6\sqrt{5}-4\sqrt{5}=2\sqrt{5}\)
Example 5 – (GRADE E)
Simplify \(\sqrt{50}+\sqrt{32}-\sqrt{128}\)
We can multiply two expressions involving surds.
Expand the brackets, then group the like terms, then simplify.
Example 6 – (GRADE E)
Expand, and simplify if possible, the following expressions
a. \(3(\sqrt{5}+2)\)
c. \((2-\sqrt{7})(6+\sqrt{7})\)
d. \((3+\sqrt{2})^2\)
END OF LESSON!
You can now answer:
EDEXCEL Pure Year 1 Ex 1E – All Questions
OCR A Student Book 1 Ex 2B – Q8 Q9 Q11 Q12 Q13* Q14*