Radical Expressions - Ninth Grade Math
Introduction to Radicals
Radical: The symbol $\sqrt{\phantom{x}}$ used to indicate a root
Radicand: The number or expression inside the radical symbol
Index: The small number indicating which root (if not shown, it's 2 for square root)
Radical Expression: An expression containing a radical symbol
Notation: $\sqrt[n]{x}$ where $n$ is the index and $x$ is the radicand
Radicand: The number or expression inside the radical symbol
Index: The small number indicating which root (if not shown, it's 2 for square root)
Radical Expression: An expression containing a radical symbol
Notation: $\sqrt[n]{x}$ where $n$ is the index and $x$ is the radicand
Basic Radical Definitions:
Square Root: $\sqrt{x}$ or $\sqrt[2]{x}$
• If $\sqrt{x} = a$, then $a^2 = x$
• Example: $\sqrt{25} = 5$ because $5^2 = 25$
Cube Root: $\sqrt[3]{x}$
• If $\sqrt[3]{x} = a$, then $a^3 = x$
• Example: $\sqrt[3]{27} = 3$ because $3^3 = 27$
nth Root: $\sqrt[n]{x}$
• If $\sqrt[n]{x} = a$, then $a^n = x$
Square Root: $\sqrt{x}$ or $\sqrt[2]{x}$
• If $\sqrt{x} = a$, then $a^2 = x$
• Example: $\sqrt{25} = 5$ because $5^2 = 25$
Cube Root: $\sqrt[3]{x}$
• If $\sqrt[3]{x} = a$, then $a^3 = x$
• Example: $\sqrt[3]{27} = 3$ because $3^3 = 27$
nth Root: $\sqrt[n]{x}$
• If $\sqrt[n]{x} = a$, then $a^n = x$
1. Simplify Radical Expressions
Simplified Radical: A radical with no perfect square factors in the radicand, no fractions under the radical, and no radicals in the denominator
Product Property of Radicals:
$$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$
where $a \geq 0$ and $b \geq 0$
Key Rule: The square root of a product equals the product of square roots
$$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$
where $a \geq 0$ and $b \geq 0$
Key Rule: The square root of a product equals the product of square roots
Steps to Simplify Radicals:
Step 1: Find the largest perfect square factor of the radicand
Step 2: Rewrite the radicand as product of perfect square and remaining factors
Step 3: Use product property to separate
Step 4: Simplify the perfect square
Step 5: Write final answer
Step 1: Find the largest perfect square factor of the radicand
Step 2: Rewrite the radicand as product of perfect square and remaining factors
Step 3: Use product property to separate
Step 4: Simplify the perfect square
Step 5: Write final answer
Perfect Squares to Memorize:
$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225$
$1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2, 11^2, 12^2, 13^2, 14^2, 15^2$
$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225$
$1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2, 11^2, 12^2, 13^2, 14^2, 15^2$
Example 1: Simplify $\sqrt{72}$
Step 1: Find largest perfect square factor
$72 = 36 \times 2$ (36 is perfect square)
Step 2: Rewrite
$\sqrt{72} = \sqrt{36 \times 2}$
Step 3: Separate
$= \sqrt{36} \cdot \sqrt{2}$
Step 4: Simplify
$= 6\sqrt{2}$
Answer: $6\sqrt{2}$
Step 1: Find largest perfect square factor
$72 = 36 \times 2$ (36 is perfect square)
Step 2: Rewrite
$\sqrt{72} = \sqrt{36 \times 2}$
Step 3: Separate
$= \sqrt{36} \cdot \sqrt{2}$
Step 4: Simplify
$= 6\sqrt{2}$
Answer: $6\sqrt{2}$
Example 2: Simplify $\sqrt{50}$
$50 = 25 \times 2$
$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$
Answer: $5\sqrt{2}$
$50 = 25 \times 2$
$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$
Answer: $5\sqrt{2}$
Example 3: Simplify $\sqrt{200}$
$200 = 100 \times 2$
$\sqrt{200} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}$
Answer: $10\sqrt{2}$
$200 = 100 \times 2$
$\sqrt{200} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}$
Answer: $10\sqrt{2}$
Example 4: Simplify $\sqrt{48}$
$48 = 16 \times 3$
$\sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$
Answer: $4\sqrt{3}$
$48 = 16 \times 3$
$\sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$
Answer: $4\sqrt{3}$
2. Simplify Radical Expressions with Variables
Rules for Variables in Radicals:
Even Powers:
$$\sqrt{x^2} = |x|$$
$$\sqrt{x^4} = x^2$$
$$\sqrt{x^{2n}} = x^n$$ (assuming $x \geq 0$)
General Rule:
$$\sqrt{x^n} = x^{n/2}$$ when $n$ is even
For Odd Powers: Break into even power plus remaining
$$\sqrt{x^5} = \sqrt{x^4 \cdot x} = x^2\sqrt{x}$$
Even Powers:
$$\sqrt{x^2} = |x|$$
$$\sqrt{x^4} = x^2$$
$$\sqrt{x^{2n}} = x^n$$ (assuming $x \geq 0$)
General Rule:
$$\sqrt{x^n} = x^{n/2}$$ when $n$ is even
For Odd Powers: Break into even power plus remaining
$$\sqrt{x^5} = \sqrt{x^4 \cdot x} = x^2\sqrt{x}$$
Steps for Variables:
Step 1: Separate numerical coefficient from variables
Step 2: Simplify numerical part
Step 3: For each variable, divide exponent by 2
Step 4: If remainder, leave under radical
Step 5: Combine results
Step 1: Separate numerical coefficient from variables
Step 2: Simplify numerical part
Step 3: For each variable, divide exponent by 2
Step 4: If remainder, leave under radical
Step 5: Combine results
Example 1: Simplify $\sqrt{x^6}$
Exponent is 6: $\frac{6}{2} = 3$
$\sqrt{x^6} = x^3$
Answer: $x^3$
Exponent is 6: $\frac{6}{2} = 3$
$\sqrt{x^6} = x^3$
Answer: $x^3$
Example 2: Simplify $\sqrt{25x^4}$
$\sqrt{25x^4} = \sqrt{25} \cdot \sqrt{x^4}$
$= 5 \cdot x^2$
$= 5x^2$
Answer: $5x^2$
$\sqrt{25x^4} = \sqrt{25} \cdot \sqrt{x^4}$
$= 5 \cdot x^2$
$= 5x^2$
Answer: $5x^2$
Example 3: Simplify $\sqrt{18x^5}$
Numerical part: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
Variable part: $\sqrt{x^5} = \sqrt{x^4 \cdot x} = x^2\sqrt{x}$
Combine: $3\sqrt{2} \cdot x^2\sqrt{x} = 3x^2\sqrt{2x}$
Answer: $3x^2\sqrt{2x}$
Numerical part: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
Variable part: $\sqrt{x^5} = \sqrt{x^4 \cdot x} = x^2\sqrt{x}$
Combine: $3\sqrt{2} \cdot x^2\sqrt{x} = 3x^2\sqrt{2x}$
Answer: $3x^2\sqrt{2x}$
Example 4: Simplify $\sqrt{48x^7y^4}$
$\sqrt{48x^7y^4} = \sqrt{16 \times 3 \times x^6 \times x \times y^4}$
$= \sqrt{16} \cdot \sqrt{x^6} \cdot \sqrt{y^4} \cdot \sqrt{3x}$
$= 4 \cdot x^3 \cdot y^2 \cdot \sqrt{3x}$
$= 4x^3y^2\sqrt{3x}$
Answer: $4x^3y^2\sqrt{3x}$
$\sqrt{48x^7y^4} = \sqrt{16 \times 3 \times x^6 \times x \times y^4}$
$= \sqrt{16} \cdot \sqrt{x^6} \cdot \sqrt{y^4} \cdot \sqrt{3x}$
$= 4 \cdot x^3 \cdot y^2 \cdot \sqrt{3x}$
$= 4x^3y^2\sqrt{3x}$
Answer: $4x^3y^2\sqrt{3x}$
3. Simplify Radical Expressions Involving Fractions
Quotient Property of Radicals:
$$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$
where $a \geq 0$ and $b > 0$
Rationalize the Denominator: Remove radicals from denominator
Multiply numerator and denominator by radical in denominator
$$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$
where $a \geq 0$ and $b > 0$
Rationalize the Denominator: Remove radicals from denominator
Multiply numerator and denominator by radical in denominator
Steps to Simplify Radical Fractions:
Step 1: Simplify numerator and denominator separately
Step 2: If radical in denominator, rationalize
Step 3: Multiply top and bottom by radical in denominator
Step 4: Simplify result
Step 5: Reduce fraction if possible
Step 1: Simplify numerator and denominator separately
Step 2: If radical in denominator, rationalize
Step 3: Multiply top and bottom by radical in denominator
Step 4: Simplify result
Step 5: Reduce fraction if possible
Example 1: Simplify $\sqrt{\frac{25}{9}}$
$\sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}$
Answer: $\frac{5}{3}$
$\sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}$
Answer: $\frac{5}{3}$
Example 2: Simplify $\frac{6}{\sqrt{3}}$
Rationalize denominator:
$\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$
Answer: $2\sqrt{3}$
Rationalize denominator:
$\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$
Answer: $2\sqrt{3}$
Example 3: Simplify $\frac{\sqrt{50}}{\sqrt{2}}$
Method 1 - Divide under one radical:
$\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5$
Method 2 - Simplify separately:
$\frac{\sqrt{50}}{\sqrt{2}} = \frac{5\sqrt{2}}{\sqrt{2}} = 5$
Answer: $5$
Method 1 - Divide under one radical:
$\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5$
Method 2 - Simplify separately:
$\frac{\sqrt{50}}{\sqrt{2}} = \frac{5\sqrt{2}}{\sqrt{2}} = 5$
Answer: $5$
Example 4: Simplify $\frac{3}{\sqrt{5}}$
$\frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$
Answer: $\frac{3\sqrt{5}}{5}$
$\frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$
Answer: $\frac{3\sqrt{5}}{5}$
4. Multiply Radical Expressions
Multiplication Rules:
Same Index:
$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$
With Coefficients:
$$m\sqrt{a} \cdot n\sqrt{b} = mn\sqrt{ab}$$
Multiply coefficients and multiply radicands separately
Same Index:
$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$
With Coefficients:
$$m\sqrt{a} \cdot n\sqrt{b} = mn\sqrt{ab}$$
Multiply coefficients and multiply radicands separately
Example 1: Multiply $\sqrt{3} \cdot \sqrt{12}$
$\sqrt{3} \cdot \sqrt{12} = \sqrt{3 \times 12} = \sqrt{36} = 6$
Answer: $6$
$\sqrt{3} \cdot \sqrt{12} = \sqrt{3 \times 12} = \sqrt{36} = 6$
Answer: $6$
Example 2: Multiply $2\sqrt{5} \cdot 3\sqrt{7}$
Multiply coefficients: $2 \times 3 = 6$
Multiply radicands: $\sqrt{5} \times \sqrt{7} = \sqrt{35}$
Result: $6\sqrt{35}$
Answer: $6\sqrt{35}$
Multiply coefficients: $2 \times 3 = 6$
Multiply radicands: $\sqrt{5} \times \sqrt{7} = \sqrt{35}$
Result: $6\sqrt{35}$
Answer: $6\sqrt{35}$
Example 3: Multiply $4\sqrt{6} \cdot 2\sqrt{3}$
Coefficients: $4 \times 2 = 8$
Radicands: $\sqrt{6 \times 3} = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
Result: $8 \times 3\sqrt{2} = 24\sqrt{2}$
Answer: $24\sqrt{2}$
Coefficients: $4 \times 2 = 8$
Radicands: $\sqrt{6 \times 3} = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
Result: $8 \times 3\sqrt{2} = 24\sqrt{2}$
Answer: $24\sqrt{2}$
Example 4: Multiply $\sqrt{x} \cdot \sqrt{x^3}$
$\sqrt{x} \cdot \sqrt{x^3} = \sqrt{x \cdot x^3} = \sqrt{x^4} = x^2$
Answer: $x^2$
$\sqrt{x} \cdot \sqrt{x^3} = \sqrt{x \cdot x^3} = \sqrt{x^4} = x^2$
Answer: $x^2$
5. Add and Subtract Radical Expressions
Like Radicals: Radicals with the same index AND the same radicand
Key Concept: Only like radicals can be combined by addition or subtraction
Key Concept: Only like radicals can be combined by addition or subtraction
Addition and Subtraction Rules:
Like Radicals:
$$a\sqrt{x} + b\sqrt{x} = (a + b)\sqrt{x}$$
$$a\sqrt{x} - b\sqrt{x} = (a - b)\sqrt{x}$$
Add/subtract coefficients, keep radical the same
Unlike Radicals: Cannot be combined!
$$\sqrt{2} + \sqrt{3} \neq \sqrt{5}$$ (common mistake!)
Like Radicals:
$$a\sqrt{x} + b\sqrt{x} = (a + b)\sqrt{x}$$
$$a\sqrt{x} - b\sqrt{x} = (a - b)\sqrt{x}$$
Add/subtract coefficients, keep radical the same
Unlike Radicals: Cannot be combined!
$$\sqrt{2} + \sqrt{3} \neq \sqrt{5}$$ (common mistake!)
Steps to Add/Subtract Radicals:
Step 1: Simplify each radical completely
Step 2: Identify like radicals
Step 3: Combine coefficients of like radicals
Step 4: Keep unlike radicals separate
Step 1: Simplify each radical completely
Step 2: Identify like radicals
Step 3: Combine coefficients of like radicals
Step 4: Keep unlike radicals separate
Example 1: Add $3\sqrt{5} + 7\sqrt{5}$
Like radicals: Both have $\sqrt{5}$
Add coefficients: $3 + 7 = 10$
Answer: $10\sqrt{5}$
Like radicals: Both have $\sqrt{5}$
Add coefficients: $3 + 7 = 10$
Answer: $10\sqrt{5}$
Example 2: Subtract $8\sqrt{3} - 2\sqrt{3}$
$(8 - 2)\sqrt{3} = 6\sqrt{3}$
Answer: $6\sqrt{3}$
$(8 - 2)\sqrt{3} = 6\sqrt{3}$
Answer: $6\sqrt{3}$
Example 3: Simplify $\sqrt{50} + \sqrt{18}$
Step 1: Simplify each
$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$
$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
Step 2: Now they're like radicals
$5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$
Answer: $8\sqrt{2}$
Step 1: Simplify each
$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$
$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
Step 2: Now they're like radicals
$5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$
Answer: $8\sqrt{2}$
Example 4: Simplify $2\sqrt{12} - \sqrt{27} + \sqrt{3}$
Simplify:
$2\sqrt{12} = 2\sqrt{4 \times 3} = 2 \cdot 2\sqrt{3} = 4\sqrt{3}$
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$
$\sqrt{3} = \sqrt{3}$
Combine:
$4\sqrt{3} - 3\sqrt{3} + \sqrt{3} = (4 - 3 + 1)\sqrt{3} = 2\sqrt{3}$
Answer: $2\sqrt{3}$
Simplify:
$2\sqrt{12} = 2\sqrt{4 \times 3} = 2 \cdot 2\sqrt{3} = 4\sqrt{3}$
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$
$\sqrt{3} = \sqrt{3}$
Combine:
$4\sqrt{3} - 3\sqrt{3} + \sqrt{3} = (4 - 3 + 1)\sqrt{3} = 2\sqrt{3}$
Answer: $2\sqrt{3}$
Example 5: Simplify $\sqrt{2} + \sqrt{8} - \sqrt{32}$
$\sqrt{2} = \sqrt{2}$
$\sqrt{8} = 2\sqrt{2}$
$\sqrt{32} = 4\sqrt{2}$
$\sqrt{2} + 2\sqrt{2} - 4\sqrt{2} = (1 + 2 - 4)\sqrt{2} = -\sqrt{2}$
Answer: $-\sqrt{2}$
$\sqrt{2} = \sqrt{2}$
$\sqrt{8} = 2\sqrt{2}$
$\sqrt{32} = 4\sqrt{2}$
$\sqrt{2} + 2\sqrt{2} - 4\sqrt{2} = (1 + 2 - 4)\sqrt{2} = -\sqrt{2}$
Answer: $-\sqrt{2}$
6. Simplify Using Distributive Property
Distributive Property with Radicals:
$$a(b + c) = ab + ac$$
With radicals:
$$\sqrt{a}(b + c) = b\sqrt{a} + c\sqrt{a}$$
$$\sqrt{a}(\sqrt{b} + \sqrt{c}) = \sqrt{ab} + \sqrt{ac}$$
$$a(b + c) = ab + ac$$
With radicals:
$$\sqrt{a}(b + c) = b\sqrt{a} + c\sqrt{a}$$
$$\sqrt{a}(\sqrt{b} + \sqrt{c}) = \sqrt{ab} + \sqrt{ac}$$
Example 1: Multiply $\sqrt{3}(2 + \sqrt{5})$
Distribute:
$\sqrt{3} \cdot 2 + \sqrt{3} \cdot \sqrt{5}$
$= 2\sqrt{3} + \sqrt{15}$
Answer: $2\sqrt{3} + \sqrt{15}$
Distribute:
$\sqrt{3} \cdot 2 + \sqrt{3} \cdot \sqrt{5}$
$= 2\sqrt{3} + \sqrt{15}$
Answer: $2\sqrt{3} + \sqrt{15}$
Example 2: Multiply $2\sqrt{5}(3\sqrt{2} - 4)$
$2\sqrt{5} \cdot 3\sqrt{2} - 2\sqrt{5} \cdot 4$
$= 6\sqrt{10} - 8\sqrt{5}$
Answer: $6\sqrt{10} - 8\sqrt{5}$
$2\sqrt{5} \cdot 3\sqrt{2} - 2\sqrt{5} \cdot 4$
$= 6\sqrt{10} - 8\sqrt{5}$
Answer: $6\sqrt{10} - 8\sqrt{5}$
Example 3: Multiply $(2 + \sqrt{3})(5 + \sqrt{3})$
Use FOIL:
F: $2 \cdot 5 = 10$
O: $2 \cdot \sqrt{3} = 2\sqrt{3}$
I: $\sqrt{3} \cdot 5 = 5\sqrt{3}$
L: $\sqrt{3} \cdot \sqrt{3} = 3$
Combine:
$10 + 2\sqrt{3} + 5\sqrt{3} + 3 = 13 + 7\sqrt{3}$
Answer: $13 + 7\sqrt{3}$
Use FOIL:
F: $2 \cdot 5 = 10$
O: $2 \cdot \sqrt{3} = 2\sqrt{3}$
I: $\sqrt{3} \cdot 5 = 5\sqrt{3}$
L: $\sqrt{3} \cdot \sqrt{3} = 3$
Combine:
$10 + 2\sqrt{3} + 5\sqrt{3} + 3 = 13 + 7\sqrt{3}$
Answer: $13 + 7\sqrt{3}$
Example 4: Multiply $(\sqrt{6} + \sqrt{2})(\sqrt{6} - \sqrt{2})$
This is difference of squares pattern!
$(a + b)(a - b) = a^2 - b^2$
$(\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4$
Answer: $4$
This is difference of squares pattern!
$(a + b)(a - b) = a^2 - b^2$
$(\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4$
Answer: $4$
7. Simplify Using Conjugates
Conjugate: A binomial formed by changing the sign between two terms
Examples:
• Conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$
• Conjugate of $\sqrt{a} - \sqrt{b}$ is $\sqrt{a} + \sqrt{b}$
Examples:
• Conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$
• Conjugate of $\sqrt{a} - \sqrt{b}$ is $\sqrt{a} + \sqrt{b}$
Conjugate Pattern (Difference of Squares):
$$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$$
$$(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$$
Key Property: Product of conjugates eliminates radicals!
Used for: Rationalizing denominators with binomials
$$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$$
$$(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$$
Key Property: Product of conjugates eliminates radicals!
Used for: Rationalizing denominators with binomials
Steps to Rationalize Using Conjugates:
Step 1: Identify the conjugate of denominator
Step 2: Multiply numerator and denominator by conjugate
Step 3: Expand numerator (use FOIL or distributive property)
Step 4: Simplify denominator (use difference of squares)
Step 5: Simplify final result
Step 1: Identify the conjugate of denominator
Step 2: Multiply numerator and denominator by conjugate
Step 3: Expand numerator (use FOIL or distributive property)
Step 4: Simplify denominator (use difference of squares)
Step 5: Simplify final result
Example 1: Rationalize $\frac{1}{\sqrt{5} + 2}$
Conjugate of denominator: $\sqrt{5} - 2$
$\frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2}$
Numerator: $1 \cdot (\sqrt{5} - 2) = \sqrt{5} - 2$
Denominator: $(\sqrt{5})^2 - 2^2 = 5 - 4 = 1$
$= \frac{\sqrt{5} - 2}{1} = \sqrt{5} - 2$
Answer: $\sqrt{5} - 2$
Conjugate of denominator: $\sqrt{5} - 2$
$\frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2}$
Numerator: $1 \cdot (\sqrt{5} - 2) = \sqrt{5} - 2$
Denominator: $(\sqrt{5})^2 - 2^2 = 5 - 4 = 1$
$= \frac{\sqrt{5} - 2}{1} = \sqrt{5} - 2$
Answer: $\sqrt{5} - 2$
Example 2: Rationalize $\frac{3}{\sqrt{7} - \sqrt{3}}$
Conjugate: $\sqrt{7} + \sqrt{3}$
$\frac{3}{\sqrt{7} - \sqrt{3}} \cdot \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}}$
Numerator: $3(\sqrt{7} + \sqrt{3}) = 3\sqrt{7} + 3\sqrt{3}$
Denominator: $(\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4$
$= \frac{3\sqrt{7} + 3\sqrt{3}}{4}$
Answer: $\frac{3\sqrt{7} + 3\sqrt{3}}{4}$
Conjugate: $\sqrt{7} + \sqrt{3}$
$\frac{3}{\sqrt{7} - \sqrt{3}} \cdot \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}}$
Numerator: $3(\sqrt{7} + \sqrt{3}) = 3\sqrt{7} + 3\sqrt{3}$
Denominator: $(\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4$
$= \frac{3\sqrt{7} + 3\sqrt{3}}{4}$
Answer: $\frac{3\sqrt{7} + 3\sqrt{3}}{4}$
Example 3: Rationalize $\frac{2 + \sqrt{3}}{1 - \sqrt{3}}$
Conjugate: $1 + \sqrt{3}$
Numerator:
$(2 + \sqrt{3})(1 + \sqrt{3})$
$= 2 + 2\sqrt{3} + \sqrt{3} + 3$
$= 5 + 3\sqrt{3}$
Denominator:
$(1)^2 - (\sqrt{3})^2 = 1 - 3 = -2$
$= \frac{5 + 3\sqrt{3}}{-2} = \frac{-5 - 3\sqrt{3}}{2}$
Answer: $\frac{-5 - 3\sqrt{3}}{2}$ or $-\frac{5 + 3\sqrt{3}}{2}$
Conjugate: $1 + \sqrt{3}$
Numerator:
$(2 + \sqrt{3})(1 + \sqrt{3})$
$= 2 + 2\sqrt{3} + \sqrt{3} + 3$
$= 5 + 3\sqrt{3}$
Denominator:
$(1)^2 - (\sqrt{3})^2 = 1 - 3 = -2$
$= \frac{5 + 3\sqrt{3}}{-2} = \frac{-5 - 3\sqrt{3}}{2}$
Answer: $\frac{-5 - 3\sqrt{3}}{2}$ or $-\frac{5 + 3\sqrt{3}}{2}$
8. Convert Between Rational Exponents and Radicals
Rational Exponent: An exponent that is a fraction
Connection: Rational exponents are another way to write radicals
Connection: Rational exponents are another way to write radicals
Rational Exponent Rules:
Basic Rule:
$$x^{1/n} = \sqrt[n]{x}$$
General Rule:
$$x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$$
where:
• $m$ = power
• $n$ = root (index)
Common Forms:
• $x^{1/2} = \sqrt{x}$ (square root)
• $x^{1/3} = \sqrt[3]{x}$ (cube root)
• $x^{2/3} = \sqrt[3]{x^2}$ or $(\sqrt[3]{x})^2$
• $x^{3/2} = \sqrt{x^3}$ or $(\sqrt{x})^3$
Basic Rule:
$$x^{1/n} = \sqrt[n]{x}$$
General Rule:
$$x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$$
where:
• $m$ = power
• $n$ = root (index)
Common Forms:
• $x^{1/2} = \sqrt{x}$ (square root)
• $x^{1/3} = \sqrt[3]{x}$ (cube root)
• $x^{2/3} = \sqrt[3]{x^2}$ or $(\sqrt[3]{x})^2$
• $x^{3/2} = \sqrt{x^3}$ or $(\sqrt{x})^3$
How to Remember:
• Denominator of exponent = Index of radical
• Numerator of exponent = Power inside or outside radical
$$x^{\frac{\text{power}}{\text{root}}} = \sqrt[\text{root}]{x^{\text{power}}}$$
• Denominator of exponent = Index of radical
• Numerator of exponent = Power inside or outside radical
$$x^{\frac{\text{power}}{\text{root}}} = \sqrt[\text{root}]{x^{\text{power}}}$$
Convert from Radical to Exponent
Example 1: Convert $\sqrt{x}$ to exponential form
Square root means index 2:
$\sqrt{x} = \sqrt[2]{x} = x^{1/2}$
Answer: $x^{1/2}$
Square root means index 2:
$\sqrt{x} = \sqrt[2]{x} = x^{1/2}$
Answer: $x^{1/2}$
Example 2: Convert $\sqrt[3]{x}$ to exponential form
$\sqrt[3]{x} = x^{1/3}$
Answer: $x^{1/3}$
$\sqrt[3]{x} = x^{1/3}$
Answer: $x^{1/3}$
Example 3: Convert $\sqrt[4]{x^3}$ to exponential form
Index is 4, power is 3:
$\sqrt[4]{x^3} = x^{3/4}$
Answer: $x^{3/4}$
Index is 4, power is 3:
$\sqrt[4]{x^3} = x^{3/4}$
Answer: $x^{3/4}$
Example 4: Convert $\sqrt{x^5}$ to exponential form
$\sqrt{x^5} = x^{5/2}$
Answer: $x^{5/2}$
$\sqrt{x^5} = x^{5/2}$
Answer: $x^{5/2}$
Convert from Exponent to Radical
Example 5: Convert $x^{1/5}$ to radical form
Denominator 5 = fifth root:
$x^{1/5} = \sqrt[5]{x}$
Answer: $\sqrt[5]{x}$
Denominator 5 = fifth root:
$x^{1/5} = \sqrt[5]{x}$
Answer: $\sqrt[5]{x}$
Example 6: Convert $x^{3/4}$ to radical form
Denominator 4 = fourth root, numerator 3 = cube:
$x^{3/4} = \sqrt[4]{x^3}$ or $(\sqrt[4]{x})^3$
Answer: $\sqrt[4]{x^3}$
Denominator 4 = fourth root, numerator 3 = cube:
$x^{3/4} = \sqrt[4]{x^3}$ or $(\sqrt[4]{x})^3$
Answer: $\sqrt[4]{x^3}$
Example 7: Convert $27^{2/3}$ and evaluate
Convert: $27^{2/3} = \sqrt[3]{27^2}$ or $(\sqrt[3]{27})^2$
Method 1: $\sqrt[3]{729} = 9$
Method 2: $(\sqrt[3]{27})^2 = 3^2 = 9$
Answer: $9$
Convert: $27^{2/3} = \sqrt[3]{27^2}$ or $(\sqrt[3]{27})^2$
Method 1: $\sqrt[3]{729} = 9$
Method 2: $(\sqrt[3]{27})^2 = 3^2 = 9$
Answer: $9$
Example 8: Convert $16^{3/2}$ and evaluate
$16^{3/2} = (\sqrt{16})^3 = 4^3 = 64$
Answer: $64$
$16^{3/2} = (\sqrt{16})^3 = 4^3 = 64$
Answer: $64$
Radical Operations Summary
| Operation | Rule | Example |
|---|---|---|
| Product | $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ | $\sqrt{2} \cdot \sqrt{8} = \sqrt{16} = 4$ |
| Quotient | $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ | $\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{25} = 5$ |
| Addition (like radicals) | $a\sqrt{x} + b\sqrt{x} = (a+b)\sqrt{x}$ | $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$ |
| Subtraction (like radicals) | $a\sqrt{x} - b\sqrt{x} = (a-b)\sqrt{x}$ | $7\sqrt{5} - 2\sqrt{5} = 5\sqrt{5}$ |
| Rationalize | Multiply by $\frac{\sqrt{b}}{\sqrt{b}}$ | $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ |
| Conjugates | $(a+\sqrt{b})(a-\sqrt{b}) = a^2-b$ | $(\sqrt{5}+2)(\sqrt{5}-2) = 5-4 = 1$ |
Rational Exponents Quick Reference
| Exponential Form | Radical Form | Description |
|---|---|---|
| $x^{1/2}$ | $\sqrt{x}$ | Square root |
| $x^{1/3}$ | $\sqrt[3]{x}$ | Cube root |
| $x^{1/n}$ | $\sqrt[n]{x}$ | nth root |
| $x^{2/3}$ | $\sqrt[3]{x^2}$ or $(\sqrt[3]{x})^2$ | Cube root, then square |
| $x^{3/2}$ | $\sqrt{x^3}$ or $(\sqrt{x})^3$ | Square root, then cube |
| $x^{m/n}$ | $\sqrt[n]{x^m}$ or $(\sqrt[n]{x})^m$ | nth root of x to the m power |
Common Mistakes to Avoid
| Wrong | Right | Explanation |
|---|---|---|
| $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ | Cannot simplify this way! | Square root of sum ≠ sum of square roots |
| $\sqrt{a - b} = \sqrt{a} - \sqrt{b}$ | Cannot simplify this way! | Square root of difference ≠ difference of roots |
| $\sqrt{9 + 16} = 3 + 4 = 7$ | $\sqrt{25} = 5$ | Add inside the radical first! |
| $2\sqrt{3} + 3\sqrt{2} = 5\sqrt{5}$ | Cannot combine | Not like radicals |
| $\sqrt{x^2} = x$ | $\sqrt{x^2} = |x|$ | Need absolute value (unless x ≥ 0) |
Success Tips for Radical Expressions:
✓ Always simplify radicals completely before combining
✓ Only like radicals (same index and radicand) can be added/subtracted
✓ Use product property: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
✓ Use quotient property: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
✓ Rationalize denominators - no radicals should remain in denominator
✓ Use conjugates for binomial denominators
✓ Memorize perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
✓ For variables: divide exponent by index to simplify
✓ Rational exponents: denominator = root, numerator = power
✓ Practice converting between radical and exponential forms!
✓ Always simplify radicals completely before combining
✓ Only like radicals (same index and radicand) can be added/subtracted
✓ Use product property: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
✓ Use quotient property: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
✓ Rationalize denominators - no radicals should remain in denominator
✓ Use conjugates for binomial denominators
✓ Memorize perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
✓ For variables: divide exponent by index to simplify
✓ Rational exponents: denominator = root, numerator = power
✓ Practice converting between radical and exponential forms!