Simplifying Radicals Index Radical Radicand Steps for Simplifying Square Roots 1. Factor the Radicand Completely or until you find a perfect root 2. Take out perfect roots (look for pairs) Note: With square roots the index is not written 3. Everything else (no pairs) stays under the radical

Root Properties: [1] [2] If you have an even index, you cannot take roots of negative numbers. Roots will be positive. [3] If you have an odd index, you can take the roots of both positive and negative numbers. Roots may be both positive and negative

General Notes: [1] 4 is the principal root [3] ±4 indicates both primary and secondary roots [2] – 4 is the secondary root (opposite of the principal root)

[C] [D] Example 1 [A] [B]

Example 2: Simplify [A] [B][C][D]

Example 3: [A] [B][C][D]

Radicals CW [1] [2] [3][4][5][6] [7] [8][9][10][11][12]

Radicals Simplifying Cube Roots (and beyond) 1. Factor the radicand completely 2. Take out perfect roots (triples) Example 1 a] b]

Example 2 a] b]

Example 3Finding Roots [A] [B] [C][D]

Example 4Applications Using Roots [A] The time T in seconds that it takes a pendulum to make a complete swing back and forth is given by the formula below, where L is the length of the pendulum in feet and g is the acceleration due to gravity. Find T for a 1.5 foot pendulum. Round to the nearest 100 th and g = 32 ft/sec 2.

Example 5Applications Using Roots [B] The distance D in miles from an observer to the horizon over flat land or water can be estimated by the formula below, where h is the height in feet of observation. How far is the horizon for a person whose eyes are at 6 feet? Round to the nearest 100 th.

Simplifying Radicals Example 1Multiplying Radical Expressions [A] [C][D] [B] 1.Multiply radicand by radicand 2.If it’s not underneath the radical then do not multiply, write together (ex: )

Example 2Foil a] b] c]d]

Example 3 Simplify Sums / Differences Find common radicand Combine like terms a]b]

Example 4Adding / Subtracting Roots [A] [B] [C][D]

Conjugate: Value that is multiplied to a radical expression That clears the radical. Rationalizing:Multiplying the denominator of a fraction by its conjugate.

Example 1Rationalizing Square Roots [A] [B]

Example 2Rationalizing Square Roots Cont’ [A] [B]

Example 3Rationalizing Cube Roots [A] [B] [C] [D]

Example 4Tougher Rationalizing [A] [B]

[1][2] [3][4] [5][6] [7][8] [9] [10] [11][12] Simplifying Radicals

Binomial Conjugate:Binomial quantity that turns the expression into a difference of squares.

Example 1Binomials Conjugates [A] [B]

Rational Exponents Property: Example 1:Rational to Radical Form A]B]C] Radicals

Example 2:Radical to Rational Form A]B]C]

Radicals CW Write in rational form Write in radical form.

Radicals Radical Equation Equation with a variable under the radical sign Extraneous Solutions Extra solutions that do not satisfy equation Radical Equation Steps [1] Isolate the radical term (if two, the more complex) [2] Square, Cube, Fourth, etc. Both Sides [3] Solve and check for extraneous solutions

Example 1Solving Radical Equations Algebraically [A][B]

Example 1 [C][D]

Radicals CW Solve Algebraically

Radicals CW Solve Algebraically

Radicals CW Solve Algebraically No Solution x = 4

Example 2Solving Graphically [A][B] x = ½

Example 2Continued [C] [D] Y = 4x = 3

Example 3No Solutions [A] [B] x = Ø

Example 4Misc. Equations [A] [B] x = 3 x = -1, -2