Introduction to Surds
Surds are irrational numbers that cannot be simplified to remove a square root (or cube root, etc.). They are roots of numbers that produce non-terminating, non-repeating decimals.
Examples of surds include:
Rational vs. Irrational:
Numbers like √4 (which equals 2) are not surds because they simplify to rational numbers.
Exact Values:
Surds provide exact values, while decimal approximations are often inexact.
Basic Rules of Surds
Fundamental Rules
The basic rules governing operations with surds:
Rule 1: Simplifying Surds
A surd can be simplified if the number under the root has a perfect square factor:
Example:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]Rule 2: Adding/Subtracting Like Surds
Like surds can be combined (similar to like terms in algebra):
Example:
\[ 3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5} \]Unlike Surds:
Cannot be combined: \( \sqrt{2} + \sqrt{3} \) remains as is.
Practice Exercise
Question 1
Simplify \( \sqrt{50} \)
\[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \]
Question 2
Simplify \( 2\sqrt{27} + 3\sqrt{3} \)
\[ 2\sqrt{27} = 2\sqrt{9 \times 3} = 6\sqrt{3} \]
\[ 6\sqrt{3} + 3\sqrt{3} = 9\sqrt{3} \]
Question 3
Simplify \( \sqrt{8} + \sqrt{18} \)
\[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]
\[ 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2} \]
Ready for more challenges?
Full Practice SetMultiplication of Surds
Multiplication Rules
The product of two surds follows specific rules that maintain mathematical consistency.
Rule 3: Multiplying Surds
The product of two surds is the surd of the product:
Examples:
\[ \sqrt{2} \times \sqrt{3} = \sqrt{6} \] \[ 2\sqrt{5} \times 3\sqrt{7} = 6\sqrt{35} \]Rule 4: Multiplying by Conjugates
Multiplying conjugate pairs eliminates the surd:
Example:
\[ (3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7 \]Practice Exercise
Question 1
Simplify \( \sqrt{3} \times \sqrt{12} \)
\[ \sqrt{3} \times \sqrt{12} = \sqrt{36} = 6 \]
Question 2
Simplify \( (2 + \sqrt{5})(2 - \sqrt{5}) \)
\[ (2 + \sqrt{5})(2 - \sqrt{5}) = 4 - 5 = -1 \]
Question 3
Simplify \( 3\sqrt{2} \times 4\sqrt{6} \)
\[ 3 \times 4 \times \sqrt{2 \times 6} = 12\sqrt{12} = 12 \times 2\sqrt{3} = 24\sqrt{3} \]
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Full Practice SetDivision of Surds
Division Rules
Division of surds follows similar principles to multiplication.
Rule 5: Dividing Surds
The quotient of two surds is the surd of the quotient:
Example:
\[ \frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3 \]Rationalization
Rationalizing Denominators
The process of eliminating surds from the denominator of a fraction.
Rule 6: Rationalizing Simple Denominators
Multiply numerator and denominator by the surd in the denominator:
Example:
\[ \frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5} \]Rule 7: Rationalizing Binomial Denominators
Multiply numerator and denominator by the conjugate of the denominator:
Example:
\[ \frac{2}{3 + \sqrt{2}} = \frac{2(3 - \sqrt{2})}{9 - 2} = \frac{6 - 2\sqrt{2}}{7} \]Practice Exercise
Question 1
Rationalize \( \frac{4}{\sqrt{3}} \)
\[ \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \]
Question 2
Rationalize \( \frac{5}{2 - \sqrt{3}} \)
\[ \frac{5}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{10 + 5\sqrt{3}}{4 - 3} = 10 + 5\sqrt{3} \]
Question 3
Simplify \( \frac{\sqrt{8}}{\sqrt{2}} \)
\[ \frac{\sqrt{8}}{\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2 \]
Ready for more challenges?
Full Practice SetMath Challenge
Timed Test in 7 Levels
Test your surds skills through progressively challenging levels