4.2 Laws of Surds
Surds are irrational numbers expressed in radical form \(\sqrt[n]{a}\) that cannot be simplified to eliminate the root, yielding infinite and non-recurring decimals. Mastering the laws of surds enables students to simplify complex expressions, perform basic arithmetic operations, rationalise denominators using conjugate surds, and solve algebraic equations involving radical terms.
Learning Objectives
- Compare rational numbers and irrational numbers, and relate surds with irrational numbers.
- Identify whether a number with a radical symbol is a surd or not a surd.
- State and apply the laws of surds: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\) and \(\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}\) for \(a > 0, b > 0\).
- Simplify expressions involving surds into the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(a\) has the largest possible value.
- Distinguish between similar surds and not similar surds.
- Perform addition, subtraction, and multiplication on expressions involving surds.
- Rationalise denominators containing surds using appropriate conjugate surds.
- Solve real-world problems and algebraic equations involving surds.
Classification of Real Numbers and Surd Rationalisation Summary
Differentiate rational numbers, irrational numbers and surds, and review the conjugate surd pairs used to rationalise denominators.
Numbers with radicals
Can the root be eliminated?
Yes: Integer / Rational (Not a surd)
Example: \(\sqrt{4}=2\), \(\sqrt[3]{27}=3\)
No: Infinite non-recurring decimal (Surd)
Example: \(\sqrt{2}\), \(\sqrt[3]{11}\)
Conjugate Surd Pairs
\(m\sqrt{a} \longleftrightarrow m\sqrt{a}\)
\(m\sqrt{a} + n\sqrt{b} \longleftrightarrow m\sqrt{a} - n\sqrt{b}\)
\(m\sqrt{a} - n\sqrt{b} \longleftrightarrow m\sqrt{a} + n\sqrt{b}\)
Short Explanation
Not all square roots or radicals are surds. A surd is an irrational root that yields a non-terminating, non-recurring decimal.
1. Rational vs Irrational Numbers
Rational numbers can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). Their decimals are terminating or recurring. Irrational numbers have infinite, non-recurring decimals.
2. Definition of Surds
Surds are numbers in root form \(\sqrt[n]{a}\), where \(a\) is a positive integer, having infinite and non-recurring decimal places. \(\sqrt[n]{a}\) is called "surd \(a\) order \(n\)".
3. Laws of Surds
For \(a > 0\) and \(b > 0\):
\(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
\(\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}\)
\((\sqrt{a})^2 = a\)
4. Simplifying Surds
Express \(\sqrt{N}\) as \(a\sqrt{b}\) by extracting the largest perfect square factor. Example: \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).
5. Similar Surds
Similar surds share the identical irrational factor. Example: \(12\sqrt{3}, 63\sqrt{3}, 25\sqrt{3}\) are similar. \(4\sqrt{12}=8\sqrt{3}\), \(5\sqrt{18}=15\sqrt{2}\), and \(5\sqrt{6}\) are not similar.
6. Operations on Surds
Addition and subtraction combine similar surds only. Multiplication distributes across brackets and simplifies using the laws of surds.
7. Rationalising Denominators
Multiply numerator and denominator by the conjugate surd. For \(\frac{1}{m\sqrt{a} + n\sqrt{b}}\), use \(m\sqrt{a} - n\sqrt{b}\), applying \((x+y)(x-y)=x^2-y^2\).
8. Solving Equations with Surds
Use substitution, rearranging to isolate the radical, squaring both sides, or factoring. Example: \(x - 4\sqrt{x} + 3 = 0\) gives \(\sqrt{x}=3\) or \(\sqrt{x}=1\), so \(x=9\) or \(x=1\).
| Expression | Conjugate Surd |
|---|---|
| \(m\sqrt{a}\) | \(m\sqrt{a}\) |
| \(m\sqrt{a} + n\sqrt{b}\) | \(m\sqrt{a} - n\sqrt{b}\) |
| \(m\sqrt{a} - n\sqrt{b}\) | \(m\sqrt{a} + n\sqrt{b}\) |
Steps for Working with Surds
Combine similar surds through addition or subtraction.
Multiply and divide using \(\sqrt{a}\sqrt{b} = \sqrt{ab}\) and \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).
Simplify into \(a\sqrt{b}\) by extracting the largest square factor.
Never leave an irrational surd in the denominator; multiply by the conjugate surd.
Multiplication Law
\(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)For \(a > 0, b > 0\)
Division Law
\(\sqrt{a} \div \sqrt{b} = \sqrt{\dfrac{a}{b}}\)For \(a > 0, b > 0\)
Try to Answer First
Answer in your mind, then press "Check Answer".
Determine whether \(\sqrt[3]{125}\) and \(\sqrt[4]{1125}\) are surds. Give your reasons.
Simplify the expression \((6 + 2\sqrt{2})(1 + 3\sqrt{2})\).
Rationalise the denominator of \(\frac{1}{7\sqrt{2} + 5\sqrt{3}}\).
Common Mistakes
- Confusing the order of a surd with scalar multiplication: thinking that \(\sqrt[n]{a}\) is equal to \(n\sqrt{a}\). Example: \(\sqrt[3]{8} = 2\), but \(3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2}\).
- Incorrectly adding surds by adding numbers under the root: writing \(\sqrt{a} + \sqrt{b} = \sqrt{a + b}\). Example: \(\sqrt{5} + \sqrt{7} = \sqrt{12}\), which is false.
- Attempting to add or subtract surds that are not similar. Example: adding \(3\sqrt{2} + 4\sqrt{3}\) directly as \(7\sqrt{5}\).
- Sign errors when rationalising denominators using conjugate surds: forgetting that \((x + y)(x - y) = x^2 - y^2\), resulting in incorrect signs in the denominator.
- Forgetting to find the largest square factor when expressing a surd in the form \(a\sqrt{b}\).
Concept Misunderstandings
All numbers written under a radical sign are surds.
Only roots that produce irrational, non-terminating, non-recurring decimals are surds. If the root simplifies to an integer or rational number (e.g., \(\sqrt{4} = 2, \sqrt{\frac{1}{4}} = 0.5, \sqrt[3]{27} = 3\)), it is not a surd.
The product of any two irrational numbers is always irrational.
The product of two irrational numbers can be rational. Example: \(\sqrt{3} \times \sqrt{3} = 3\), or \((1 - \sqrt{2})(1 + \sqrt{2}) = 1 - 2 = -1\).
Recurring decimals are irrational because they have infinite decimal places.
Recurring decimals (like \(0.3333\dots = \frac{1}{3}\) or \(0.6767\dots = \frac{67}{99}\)) can always be converted into fractions of integers, so they are rational numbers.
Summary
- Surds are irrational roots \(\sqrt[n]{a}\) with infinite, non-recurring decimals.
- Multiplication law: \(\sqrt{a}\sqrt{b} = \sqrt{ab}\) and division law: \(\sqrt{a}/\sqrt{b} = \sqrt{a/b}\) for \(a, b > 0\).
- Surds can be simplified into \(a\sqrt{b}\) by extracting the largest square factor.
- Only similar surds (sharing the same radical) can be added or subtracted.
- Denominators with surds are rationalised by multiplying both numerator and denominator by the conjugate surd.
Short Activity
Objective: Identify surds, conjugate surds, laws of surds and simplified forms.
A. Objective Quiz
1 Which of the following expressions is a surd?
2 What is the conjugate surd of the expression \(2\sqrt{3} - 5\sqrt{7}\)?
B. Fill in the Blanks
3 For \(a > 0\) and \(b > 0\), the first law of surds states that \(\sqrt{a} \times \sqrt{b} =\) ________.
4 When \(\sqrt{18}\) is written in the form \(a\sqrt{b}\) where \(a\) is the largest integer, the value of \(a\) is ________.
C. Matching / Drag and Drop
Match each radical expression with its correct simplified form or classification. On a phone, tap a card first, then tap the matching box.
