Solving Radicals Step 1: Get rid of anything that might not be under the radical. Step 2: Square both sides to get rid of the radical. Step 3: Isolate the variable. Step 4: Solve Step 5: Plug answer back into equation and make sure that it works.
What is a Radical Expression? A Radical Expression is an equation that has a variable in a radicand or has a variable with a rational exponent. yes no
EXAMPLE – Solving a Radical Equation 2 () 2 Square both sides to get rid of the square root
EXAMPLE 2 () 2 NO SOLUTION Since 16 doesn’t plug in as a solution. Let’s Double Check that this works Note: You will get Extraneous Solutions from time to time – always do a quick check
Can graphing calculators help? SURE! 1.Input for Y1 2.Input x-2 for Y2 3.Graph 4.Find the points of intersection One Solution at (4, 2) To see if this is extraneous or not, plug the x value back into the equation. Does it work?
An Arithmetic Sequence is defined as a sequence in which there is a common difference between consecutive terms.
Which of the following sequences are arithmetic? Identify the common difference. YES YES YES NO NO
T h e c o m m o n d i f f e r e n c e i s a l w a y s t h e d i f f e r e n c e b e t w e e n a n y t e r m a n d t h e t e r m t h a t p r o c e e d s t h a t t e r m. C o m m o n D i f f e r e n c e = 5
The general form of an ARITHMETIC sequence. First Term: Second Term: Third Term: Fourth Term: Fifth Term: nth Term:
Formula for the nth term of an ARITHMETIC sequence. I f w e k n o w a n y t h r e e o f t h e s e w e o u g h t t o b e a b l e t o f i n d t h e f o u r t h.
Introduction Geometric sequences are exponential functions that have a domain of consecutive positive integers. Geometric sequences can be represented by formulas, either explicit or recursive, and those formulas can be used to find a certain term of the sequence or the number of a certain value in the sequence. 42 3.8.2: Geometric Sequences
Guided Practice Example 1 Find the constant ratio, write the explicit formula, and find the seventh term for the following geometric sequence. 3, 1.5, 0.75, 0.375, … 43 3.8.2: Geometric Sequences
Guided Practice: Example 1, continued 1.Find the constant ratio by dividing two successive terms. 1.5 ÷ 3 = 0.5 44 3.8.2: Geometric Sequences
Guided Practice: Example 1, continued 2.Confirm that the ratio is the same between all of the terms. 0.75 ÷ 1.5 = 0.5 and 0.375 ÷ 0.75 = 0.5 45 3.8.2: Geometric Sequences
Guided Practice: Example 1, continued 3.Identify the first term (a 1 ). a 1 = 3 46 3.8.2: Geometric Sequences
Guided Practice: Example 1, continued 4.Write the explicit formula. a n = a 1 r n – 1 Explicit formula for any given geometric sequence a n = (3)(0.5) n – 1 Substitute values for a 1 and n. 47 3.8.2: Geometric Sequences
Guided Practice: Example 1, continued 5.To find the seventh term, substitute 7 for n. a 7 = (3)(0.5) 7 – 1 a 7 = (3)(0.5) 6 Simplify. a 7 = 0.046875Multiply. The seventh term in the sequence is 0.046875. 48 3.8.2: Geometric Sequences ✔
Guided Practice Example 3 A geometric sequence is defined recursively by, with a 1 = 729. Find the first five terms of the sequence, write an explicit formula to represent the sequence, and find the eighth term. 49 3.8.2: Geometric Sequences
Guided Practice: Example 3, continued 1.Using the recursive formula: 50 3.8.2: Geometric Sequences
Guided Practice: Example 3, continued The first five terms of the sequence are 729, –243, 81, –27, and 9. 51 3.8.2: Geometric Sequences
Guided Practice: Example 3, continued 2.The first term is a 1 = 729 and the constant ratio is, so the explicit formula is. 52 3.8.2: Geometric Sequences
Guided Practice: Example 3, continued 3.Substitute 8 in for n and evaluate. The eighth term in the sequence is. 53 3.8.2: Geometric Sequences ✔