Definition of Surds
A surd is an irrational number expressed as a root (√) that cannot be simplified to remove the root. Surds are roots of numbers that don't result in whole numbers.
Examples of surds:
\[ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt[3]{7}, 1 + \sqrt{2} \]Non-examples (not surds):
\[ \sqrt{4} = 2, \sqrt[3]{27} = 3 \]Pure Surd:
Contains only irrational terms
\[ \sqrt{3}, 2\sqrt{5} \]Mixed Surd:
Contains both rational and irrational terms
\[ 3 + \sqrt{2}, 5 - 2\sqrt{3} \]Laws of Surds
Fundamental rules for working with surds:
- \[ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \]
- \[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \]
- \[ (\sqrt{a})^n = \sqrt{a^n} \]
- \[ \sqrt[n]{a} = a^{1/n} \]
Simplifying Surds
Express the surd in its simplest form by factoring out perfect squares:
Simplify \( \sqrt{50} \):
\[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \]Simplify \( \sqrt{72} \):
Operations with Surds
Addition and Subtraction
Can only add/subtract like surds (same irrational part):
Example 1:
\[ 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \]Example 2:
\[ 4\sqrt{3} - \sqrt{3} = 3\sqrt{3} \]Cannot simplify:
\[ \sqrt{2} + \sqrt{3} \text{ remains } \sqrt{2} + \sqrt{3} \]Simplify \( 2\sqrt{5} + 3\sqrt{5} - \sqrt{5} \):
Multiplication
Multiply coefficients with coefficients and surds with surds:
Example 1:
\[ 3\sqrt{2} \times 2\sqrt{5} = 6\sqrt{10} \]Example 2:
\[ \sqrt{3} \times \sqrt{12} = \sqrt{36} = 6 \]Simplify \( (2\sqrt{3})(5\sqrt{6}) \):
Division
Divide coefficients with coefficients and surds with surds:
Example:
\[ \frac{6\sqrt{10}}{2\sqrt{5}} = 3\sqrt{2} \]Simplify \( \frac{12\sqrt{15}}{3\sqrt{3}} \):
Rationalization
Rationalizing means eliminating surds from the denominator of a fraction by multiplying numerator and denominator by the conjugate or appropriate surd.
Simple Denominators
Multiply numerator and denominator by the surd in the denominator:
Rationalize \( \frac{3}{\sqrt{5}} \):
\[ \frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5} \]Binomial Denominators
Use the conjugate (change the sign between terms) to rationalize:
Rationalize \( \frac{2}{3 + \sqrt{2}} \):
\[ \frac{2}{3 + \sqrt{2}} \times \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{6 - 2\sqrt{2}}{9 - 2} = \frac{6 - 2\sqrt{2}}{7} \]Conjugate Pairs:
For \( a + \sqrt{b} \), conjugate is \( a - \sqrt{b} \)
For \( \sqrt{a} + \sqrt{b} \), conjugate is \( \sqrt{a} - \sqrt{b} \)
Rationalize \( \frac{5}{2 - \sqrt{3}} \):
Equal Surds
Two surds are equal if their simplified forms are identical, or if their rational and irrational parts are equal.
Example 1:
\[ \sqrt{12} = 2\sqrt{3} \text{ because both simplify to the same value} \]Example 2:
\[ a + b\sqrt{2} = 3 + 5\sqrt{2} \implies a = 3 \text{ and } b = 5 \]Find x and y if \( (x + y) + (x - y)\sqrt{3} = 4 + 2\sqrt{3} \):
Set up equations:
\[ x + y = 4 \] \[ x - y = 2 \]Solving gives \( x = 3 \), \( y = 1 \)
Surd Equations
Equations containing surds can be solved by isolating the surd and squaring both sides.
Solve \( \sqrt{2x + 3} = 5 \):
\[ (\sqrt{2x + 3})^2 = 5^2 \] \[ 2x + 3 = 25 \] \[ 2x = 22 \] \[ x = 11 \]Important Note
Always check solutions in the original equation as squaring can introduce extraneous solutions.
Solve \( \sqrt{x - 1} = x - 3 \):
Square both sides:
\[ x - 1 = (x - 3)^2 \] \[ x - 1 = x^2 - 6x + 9 \] \[ 0 = x^2 - 7x + 10 \] \[ (x - 5)(x - 2) = 0 \]Solutions: x = 5 or x = 2
Check x=5: \( \sqrt{4} = 2 \) ✓
Check x=2: \( \sqrt{1} = -1 \) ✗ (extraneous)
Final answer: x = 5
Equations with Multiple Surds
Isolate one surd before squaring both sides. May need to square twice.
Solve \( \sqrt{x + 5} - \sqrt{x - 3} = 2 \):
- Isolate one surd: \( \sqrt{x + 5} = 2 + \sqrt{x - 3} \)
- Square both sides: \( x + 5 = 4 + 4\sqrt{x - 3} + x - 3 \)
- Simplify: \( 4 = 4\sqrt{x - 3} \) → \( 1 = \sqrt{x - 3} \)
- Square again: \( 1 = x - 3 \) → \( x = 4 \)
- Check: \( \sqrt{9} - \sqrt{1} = 3 - 1 = 2 \) ✓
Practice Exercise
Question 1
Simplify \( \sqrt{27} + \sqrt{75} - \sqrt{48} \).
Question 2
Rationalize the denominator: \( \frac{4}{1 - \sqrt{3}} \).
Question 3
Solve the equation: \( \sqrt{3x + 4} = x \).
Square both sides:
\[ 3x + 4 = x^2 \] \[ x^2 - 3x - 4 = 0 \] \[ (x - 4)(x + 1) = 0 \]x = 4 or x = -1
Check x=4: \( \sqrt{16} = 4 \) ✓
Check x=-1: \( \sqrt{1} = -1 \) ✗
Solution: x = 4
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