Radical Expressions and Functions
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Simplify Radical Expressions with Variables
Product Property of Radicals:
\[ \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \]
This property allows us to break down radicals into simpler factors
Steps to Simplify:
1. Factor the radicand (number/expression under the radical)
2. Identify perfect powers that match the index
3. Bring perfect powers out of the radical
4. Use absolute value for even roots with variables (when necessary)
Key Rules:
For Square Roots:
• \( \sqrt{x^2} = |x| \) (absolute value for even roots)
• \( \sqrt{x^{2n}} = x^n \) when x is known to be positive
• Bring out pairs of factors
General Rule:
\[ \sqrt[n]{x^m} = x^{m/n} \]
Examples:
Simplify: \( \sqrt{50x^4y^3} \)
Factor: \( \sqrt{25 \cdot 2 \cdot x^4 \cdot y^2 \cdot y} \)
Identify perfect squares: \( \sqrt{25} \cdot \sqrt{x^4} \cdot \sqrt{y^2} \cdot \sqrt{2y} \)
Answer: \( 5x^2y\sqrt{2y} \)
Simplify: \( \sqrt{72a^5b^6} \)
Factor: \( \sqrt{36 \cdot 2 \cdot a^4 \cdot a \cdot b^6} \)
Simplify: \( 6a^2b^3\sqrt{2a} \)
Answer: \( 6a^2b^3\sqrt{2a} \)
Simplify: \( \sqrt[3]{16x^7y^4} \)
Factor: \( \sqrt[3]{8 \cdot 2 \cdot x^6 \cdot x \cdot y^3 \cdot y} \)
Identify perfect cubes: \( 2x^2y\sqrt[3]{2xy} \)
Answer: \( 2x^2y\sqrt[3]{2xy} \)
2. Nth Roots
Definition:
The nth root of a number x is a number r that, when raised to the power n, equals x
\[ \sqrt[n]{x} = r \quad \text{means} \quad r^n = x \]
where n is called the index (or order) of the root
Properties of Nth Roots:
| Property | Formula |
|---|---|
| Product Property | \( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \) |
| Quotient Property | \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \) |
| Power Property | \( \sqrt[n]{a^m} = a^{m/n} \) |
| Root of Root | \( \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} \) |
Special Cases:
Even Roots (n is even):
• \( \sqrt[n]{x} \) is defined only when \( x \geq 0 \)
• Result is always non-negative (principal root)
Odd Roots (n is odd):
• \( \sqrt[n]{x} \) is defined for all real x
• Sign of result matches sign of x
Examples:
Evaluate: \( \sqrt[4]{81} \)
Find r such that \( r^4 = 81 \)
\( 3^4 = 81 \)
Answer: 3
Evaluate: \( \sqrt[3]{-27} \)
Find r such that \( r^3 = -27 \)
\( (-3)^3 = -27 \)
Answer: -3
Simplify: \( \sqrt[6]{x^3} \)
Use power property: \( x^{3/6} = x^{1/2} \)
Answer: \( \sqrt{x} \) or \( x^{1/2} \)
3. Domain and Range of Radical Functions
General Forms:
Even Index (Square Root, 4th root, etc.):
\[ f(x) = \sqrt[n]{ax + b} + k \quad \text{(n even)} \]
Finding Domain:
Set radicand ≥ 0: \( ax + b \geq 0 \)
Solve for x to get domain
Range:
\( [k, \infty) \) (since \( \sqrt[n]{x} \geq 0 \) for even n)
Odd Index (Cube Root, 5th root, etc.):
\[ f(x) = \sqrt[n]{ax + b} + k \quad \text{(n odd)} \]
Domain:
All real numbers: \( (-\infty, \infty) \)
Range:
All real numbers: \( (-\infty, \infty) \)
Examples:
Find domain and range: \( f(x) = \sqrt{x - 5} \)
Set radicand ≥ 0: \( x - 5 \geq 0 \)
Solve: \( x \geq 5 \)
Domain: \( [5, \infty) \)
Range: \( [0, \infty) \)
Find domain and range: \( g(x) = \sqrt{2x + 6} - 3 \)
Set radicand ≥ 0: \( 2x + 6 \geq 0 \)
Solve: \( 2x \geq -6 \) → \( x \geq -3 \)
Domain: \( [-3, \infty) \)
Range: \( [-3, \infty) \) (shifted down 3 units)
Find domain and range: \( h(x) = \sqrt[3]{x + 2} \)
Odd index (3): defined for all real numbers
Domain: \( (-\infty, \infty) \)
Range: \( (-\infty, \infty) \)
4. Graph Square Root Functions
Parent Function \( f(x) = \sqrt{x} \):
Key Features:
• Domain: \( [0, \infty) \)
• Range: \( [0, \infty) \)
• Starting Point: (0, 0)
• Direction: Increasing
• Shape: Half of a parabola on its side
Key Points for Parent Function:
| x | \( f(x) = \sqrt{x} \) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
Transformations:
\[ f(x) = a\sqrt{x - h} + k \]
• h: Horizontal shift (right if h > 0, left if h < 0)
• k: Vertical shift (up if k > 0, down if k < 0)
• a: Vertical stretch if |a| > 1, compression if 0 < |a| < 1
• a < 0: Reflection over x-axis
Graphing Steps:
1. Find the starting point (h, k)
2. Determine domain: x ≥ h
3. Plot key points by choosing x-values ≥ h
4. Draw smooth curve starting at (h, k)
5. Curve increases gradually to the right
Example:
Graph: \( f(x) = \sqrt{x - 2} + 1 \)
Starting point: (2, 1) (shifted right 2, up 1)
Domain: \( [2, \infty) \)
Range: \( [1, \infty) \)
Key points: (2,1), (3,2), (6,3), (11,4)
5. Solve Radical Equations
Method:
Steps to Solve:
1. Isolate the radical on one side
2. Raise both sides to the power that matches the index
3. Solve the resulting equation
4. CHECK ALL SOLUTIONS in the original equation
5. Reject extraneous solutions (solutions that don't work)
Extraneous Solutions:
When you square (or raise to any even power) both sides of an equation, you may introduce solutions that don't satisfy the original equation
⚠️ Always Check Your Solutions!
Substitute each solution back into the original equation to verify it works
Examples:
Solve: \( \sqrt{x + 3} = 5 \)
Square both sides: \( (\sqrt{x+3})^2 = 5^2 \)
Simplify: \( x + 3 = 25 \)
Solve: \( x = 22 \)
Check: \( \sqrt{22 + 3} = \sqrt{25} = 5 \) ✓
Solution: x = 22
Solve: \( \sqrt{2x - 1} = x - 2 \)
Square both sides: \( 2x - 1 = (x - 2)^2 \)
Expand: \( 2x - 1 = x^2 - 4x + 4 \)
Rearrange: \( 0 = x^2 - 6x + 5 \)
Factor: \( (x - 5)(x - 1) = 0 \)
Potential solutions: x = 5 or x = 1
Check x = 5: \( \sqrt{9} = 3 \) ✓
Check x = 1: \( \sqrt{1} = 1 \neq -1 \) ✗ (Extraneous)
Solution: x = 5 only
Solve: \( \sqrt{x + 7} - 2 = x \)
Isolate radical: \( \sqrt{x + 7} = x + 2 \)
Square both sides: \( x + 7 = (x + 2)^2 \)
Expand: \( x + 7 = x^2 + 4x + 4 \)
Rearrange: \( 0 = x^2 + 3x - 3 \)
Use quadratic formula: \( x = \frac{-3 \pm \sqrt{9 + 12}}{2} = \frac{-3 \pm \sqrt{21}}{2} \)
\( x \approx 0.79 \) or \( x \approx -3.79 \)
Check both in original equation
Solution: \( x = \frac{-3 + \sqrt{21}}{2} \) only
Solve: \( \sqrt[3]{2x + 5} = 3 \)
Cube both sides: \( (\sqrt[3]{2x+5})^3 = 3^3 \)
Simplify: \( 2x + 5 = 27 \)
Solve: \( 2x = 22 \) → \( x = 11 \)
Check: \( \sqrt[3]{27} = 3 \) ✓
Solution: x = 11
6. Quick Reference Summary
Key Formulas & Rules:
Simplifying: \( \sqrt[n]{x^m} = x^{m/n} \)
Product: \( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \)
Quotient: \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \)
Domain (even index): Set radicand ≥ 0
Domain (odd index): All real numbers
Solving: Isolate radical, raise to power, CHECK solutions!
📚 Study Tips
✓ Look for perfect powers when simplifying radicals
✓ Even index requires non-negative radicand
✓ Odd index works for all real numbers
✓ Always check for extraneous solutions when solving
✓ Starting point of √(x-h)+k is at (h, k)