IB MYP 4-5 Maths- Surds, roots and radicals, including simplifying- Study Notes - New Syllabus
IB MYP 4-5 Maths- Surds, roots and radicals, including simplifying – Study Notes
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- Surds, roots and radicals, including simplifying
IB MYP 4-5 Maths- Surds, roots and radicals, including simplifying – Study Notes – All topics
Roots and Radicals
Roots and Radicals
A radical is an expression that includes a root symbol (√). The most common radical is the square root \( \sqrt{x} \), but cube roots \( \sqrt[3]{x} \), fourth roots, etc. also exist.
| Mathematical Rule | Explanation |
|---|---|
| \( \sqrt{a^2} = |a| \) | Square root of a square gives the absolute value |
| \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) | You can split roots across multiplication |
| \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) | You can combine roots when dividing |
| \( \sqrt[n]{x^m} = x^{\frac{m}{n}} \) | Roots can be expressed using fractional powers |
- Radicand: The value inside the radical (e.g. in \( \sqrt{25} \), the radicand is 25).
- Index: The small number indicating the type of root (e.g. 2 for square root, 3 for cube root).
- Square root: Means “what number squared gives this value?”
Always simplify roots where possible .
Only square roots of perfect squares (like 1, 4, 9, 16, 25…) result in whole numbers.
Example:
Write \( \sqrt[3]{27} \) as a whole number and \( \sqrt{25} \) as a power.
▶️ Answer/Explanation
Simplify roots
\( \sqrt[3]{27} = 3 \) because \( 3^3 = 27 \)
\( \sqrt{25} = 5 = 25^{\frac{1}{2}} \)
Example:
Simplify the following expression:
$\left( 81^{\frac{1}{4}} \cdot \sqrt{50} \right) \div \sqrt{2}$
▶️ Answer/Explanation
Convert fractional exponent to root
\( 81^{\frac{1}{4}} = \sqrt[4]{81} = 3 \) (because \( 3^4 = 81 \))
Simplify \( \sqrt{50} \)
\( \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \)
Multiply the simplified parts
\( 3 \cdot 5\sqrt{2} = 15\sqrt{2} \)
Divide by \( \sqrt{2} \)
\( \frac{15\sqrt{2}}{\sqrt{2}} = 15 \)
Surds
Surd
A surd is a square root (or cube root, etc.) that cannot be simplified to a rational number. Surds are irrational numbers written in root form.
Characteristics of a Surd:
- It cannot be written as a fraction \( \frac{a}{b} \).
- Its decimal form is non-terminating and non-repeating.
- It is left in root form to express the exact value.
Examples of Surds:
- \( \sqrt{2},\ \sqrt{3},\ \sqrt{5},\ \sqrt{7} \)
- \( \sqrt{11},\ \sqrt{13},\ \sqrt{17},\ \sqrt{19} \)
- \( \sqrt[3]{2},\ \sqrt[3]{5},\ \sqrt[4]{7} \)
These values cannot be written exactly as decimals or fractions, so we leave them as surds.
Not Surds (Because They Simplify):
- \( \sqrt{4} = 2 \)
- \( \sqrt{9} = 3 \)
- \( \sqrt{16} = 4 \)
- \( \sqrt{\frac{1}{4}} = \frac{1}{2} \)
These are not surds because they simplify to rational numbers.
Example:
Identify which of the following are surds:
\( \sqrt{2},\ \sqrt{25},\ \sqrt{3},\ \frac{1}{2},\ \sqrt{7} \)
▶️ Answer/Explanation
Simplify each root
\( \sqrt{2} \) – cannot be simplified → surd
\( \sqrt{25} = 5 \) – rational → not a surd
\( \sqrt{3} \) – cannot be simplified → surd
\( \frac{1}{2} \) – rational → not a surd
\( \sqrt{7} \) – cannot be simplified → surd
Simplifying Surds
To simplify a surd: Find square factors inside the root.
Use the rule: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \)
- \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \)
- \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \)
Example:
Simplify: \( \sqrt{72} \)
▶️ Answer/Explanation
Break 72 into square × other
\( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \)
Example:
Simplify: \( \sqrt{98} \)
▶️ Answer/Explanation
Break 98 into a square × another number
\( \sqrt{98} = \sqrt{49 \cdot 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2} \)
Adding and Subtracting Surds
Rule: You can only add or subtract like surds (same root).
- \( 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \)
- \( 7\sqrt{3} – 2\sqrt{3} = 5\sqrt{3} \)
- \( 2\sqrt{5} + 3\sqrt{7} \) cannot be simplified
Example:
Simplify: \( 4\sqrt{3} + 2\sqrt{12} \)
▶️ Answer/Explanation
Simplify \( \sqrt{12} \)
\( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \)
Now add like terms
\( 4\sqrt{3} + 2(2\sqrt{3}) = 4\sqrt{3} + 4\sqrt{3} = 8\sqrt{3} \)
Example:
Simplify: \( 3\sqrt{5} + \sqrt{20} – \sqrt{45} \)
▶️ Answer/Explanation
Simplify each surd first
\( \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \)
\( \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \)
Now combine like terms:
\( 3\sqrt{5} + 2\sqrt{5} – 3\sqrt{5} = 2\sqrt{5} \)
Multiplying and Dividing Surds
Multiplication Rule: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \)
Division Rule: \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \)
Example:
Simplify: \( \sqrt{5} \cdot \sqrt{20} \)
▶️ Answer/Explanation
Multiply under the root
\( \sqrt{5 \cdot 20} = \sqrt{100} = 10 \)
Example:
Simplify: \( \frac{\sqrt{48}}{\sqrt{3}} \)
▶️ Answer/Explanation
Combine under one root
\( \frac{\sqrt{48}}{\sqrt{3}} = \sqrt{\frac{48}{3}} = \sqrt{16} = 4 \)
Rationalizing the Denominator
Why: Do not leave a root in the denominator.
Rule: Multiply numerator and denominator by the surd.
Formula: \( \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} \)
Example:
Rationalize: \( \frac{5}{\sqrt{3}} \)
▶️ Answer/Explanation
Multiply numerator and denominator by \( \sqrt{3} \)
\( \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3} \)
Example:
Rationalize the denominator: $ \frac{5}{2 + \sqrt{3}}$
▶️ Answer/Explanation
Multiply numerator and denominator by the conjugate of the denominator
Conjugate of \( 2 + \sqrt{3} \) is \( 2 – \sqrt{3} \)
Multiply using difference of squares
Numerator: \( 5(2 – \sqrt{3}) = 10 – 5\sqrt{3} \)
Denominator: \( (2 + \sqrt{3})(2 – \sqrt{3}) = 4 – 3 = 1 \)
Simplify
\( \frac{10 – 5\sqrt{3}}{1} = 10 – 5\sqrt{3} \)
Surds in Algebra
You may have to expand and simplify algebraic expressions involving surds.
- \( (2 + \sqrt{3})(2 – \sqrt{3}) \) – Use difference of squares
- \( (x + \sqrt{5})^2 = x^2 + 2x\sqrt{5} + 5 \)
Example:
Expand and simplify: \( (3 + \sqrt{2})(3 – \sqrt{2}) \)
▶️ Answer/Explanation
Use identity: \( (a + b)(a – b) = a^2 – b^2 \)
\( 3^2 – (\sqrt{2})^2 = 9 – 2 = 7 \)
Example:
Expand and simplify: \( (2\sqrt{3} + \sqrt{12})(\sqrt{3} – \sqrt{27}) \)
▶️ Answer/Explanation
Simplify surds first:
\( \sqrt{12} = 2\sqrt{3} \), \( \sqrt{27} = 3\sqrt{3} \)
Now the expression becomes:
\( (2\sqrt{3} + 2\sqrt{3})(\sqrt{3} – 3\sqrt{3}) = 4\sqrt{3} \cdot (-2\sqrt{3}) \)
Multiply:
\( 4\sqrt{3} \cdot -2\sqrt{3} = -8(\sqrt{3})^2 = -8 \cdot 3 = -24 \)
Example:
Simplify the following expression fully:
$ \frac{2\sqrt{18} + \sqrt{50}}{\sqrt{2}} + (3 + \sqrt{3})^2$
▶️ Answer/Explanation
Simplify all surds
\( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \)
\( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \)
Substitute simplified surds into expression
\( \frac{2(3\sqrt{2}) + 5\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2} + 5\sqrt{2}}{\sqrt{2}} = \frac{11\sqrt{2}}{\sqrt{2}} = 11 \)
Expand the squared binomial
\( (3 + \sqrt{3})^2 = 3^2 + 2 \cdot 3 \cdot \sqrt{3} + (\sqrt{3})^2 = 9 + 6\sqrt{3} + 3 = 12 + 6\sqrt{3} \)
Add all parts together
\( 11 + 12 + 6\sqrt{3} = 23 + 6\sqrt{3} \)