Simplifying Radicals Index Radical Radicand Note With square

Simplifying Radicals Index Radical Radicand Note: With square roots the index is not written 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) 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] [2] [3] 4 is the principal root – 4 is the secondary root (opposite of the principal root) ± 4 indicates both primary and secondary roots

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

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

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

Radicals CW Solutions [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 3 Finding Roots [A] [B] [C] [D]

Example 4 [A] Applications Using Roots 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 5 [B] Applications Using Roots 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 1. Multiply radicand by radicand 2. If it’s not underneath the radical then do not multiply, write together (ex: ) Example 1 Multiplying Radical Expressions [A] [B] [C] [D]

Example 2 Foil a] b] c] d]

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

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

SPECIAL FRACTION EXPONENT: The exponent is most often used in the power of monomials. Examples: Do you notice any other type of mathematical symbols that these special fraction exponents represent?

Special Fraction Exponents, , are more commonly known as radicals in which the N value represents the root or index of Index the radical. Radical Symbol Radicals: Radicand Note: The square root or ½ exponent is the most common radical and does not need to have the index written. Steps for Simplifying Square Roots 1. Prime Factorization: Factor the Radicand Completely 2. Write the base of all perfect squares (PAIRS) outside of the radical as product 3. Everything else (SINGLES) stays under the radical as a product.

Operations with Rational (Fraction) Exponents The same operations of when to multiply, add, subtract exponents apply with rational (fraction) exponents as did with integer (whole) exponents • Hint: Remember how to find common denominators and reduce. 1) 4) 2) 5) 3) 6)

Radicals CW Write in rational form. 1. 2. 3. 4. 7. 8. Write in radical form. 5. 6.

Radicals (Roots) and Rational Exponent Form Rational Exponents Property: OR OR Example 1: Change Rational to Radical Form A] Example 2: A] B] C] Change Radical to Rational Form B] C]

Radicals Classwork # 1 – 4: Write in rational form. 1. 2. 3. 4. 7. 8. #5 – 8: Write in radical form. 5. 6.

Radicals Classwork #2 Determine if each pair are equivalent statements or not. 1. 3. 5. and and 2. 4. 6. and and

Simplifying Rational Exponents • Apply normal operations with exponents. • Convert to radical form. • Simplify the radical expression based on the index and radicand. 1. 2. 3. 4. 5. 6. 7. 8.

Radicals Classwork #3 Simplify the following expressions into simplest radical form 1. 4. 2. 5. 3. 6.

Change of Base (Index or Root) • Write the radicand in prime factorization form • REDUCE the fractions of Rational Exponents to rewrite radicals. 1. 2. 3. 4. 3.

Change of Base Practice Problems 1. 4. 2. 5. 3. 6.

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

Example 1 [C] [D]

Radicals CW Solve Algebraically. 9. 10. 11. 12.

Radicals CW Solve Algebraically. 13. 14.

Radicals CW Solve Algebraically. 15. 16. No Solution x=4

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

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

Example 3 [A] No Solutions [B] x=Ø

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