The trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction.

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1 Complex Fractions A complex fraction is an expression that features fractions within fractions. To simplify complex fractions, we only need to master one very simple method. Simplify The trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction. The little fractions have denominators 6, 8, and 4. The LCM of 6, 8, and 4 is 24. So, multiply the top and bottom by 24: = = = There really is nothing more to it. No matter how complicated the complex fractions may look, this method will work! Just remember: If you are dealing with a variable in the denominator, be sure to note excluded values (values of the variable that would make the denominator zero).

2 Practice: simplify complex fractions 1. 1 x 1 y 1 x 2 1 (Note the excluded values x 0 and y 0) y x x x x 3 (Factor (x 2 9), then find the LCD) x+3

3 Radical Equations A radical equation is an equation in which at least one variable is stuck inside of a radical sign (usually, a square root). For example: x + 1 = 2 To solve radical equations, we isolate the variable (as usual). Solve x + 1 = 2 To undo the square root, square both sides of the equation (in other words, raise both sides of the equation to a power of 2): We get: ( x + 1) 2 = (2) 2 x + 1 = 4 Then, subtract 1 from both sides of the equation: We get: x = 4 1 x = 3

4 In the previous example, the radical was isolated on the left-hand side to begin with. This is often not the case! If the radical is not isolated to begin with, don t jump straight to undoing the radical! Solve x + 2 = 6 We want to isolate x. Start by subtracting 2 from both sides of the equation: We get: x = 6 2 x = 4 Now, once we have the radical isolated, we can square both sides of the equation to undo the square root: x = 16

5 Check your Solutions! 4 Keep in mind that even roots (for example, x, x, x, etc.) are always positive. Test makers like to set up false equations that, even if you perform all of your algebra correctly, will yield incorrect solutions! Solve x + 1 = 3 32 Subtract 1 from both sides of the equation: x = 3 1 We get: x = 4 *TRAP* If we were to square both sides, we would get x = 16, but this answer is incorrect! Once we see x = 4, we know there is actually no solution, since even roots are always positive! Thus, the correct answer to this problem is: NO SOLUTION

6 Fractional Exponents Radical expressions can be written using radical signs or fractional exponents. For example: x can be written as x x can be written as x x 5 can be written as x Solve x 8 4 = 1 Add 4 to both sides of the equation: We get: Rewrite the radical as an exponent: 3 x = x 8 = 5 x 8 3 = 5 Now, to solve for x, we have to undo the fractional exponent. Raise both sides of the equation by a power of 3 8, the reciprocal of 8 3. Simplifying: 3 (x ) = 5 8 x = *NOTE: It may be tempting to approximate the solution to a decimal. DO NOT APPROXIMATE unless you are asked to do so is the exact answer.

7 Practice: solving radical equations 5 1. Solve x 1 = Solve x = 1 3. Solve 2 a 1 = 3 4. Solve x = 3 5. Solve 2x = 3

8 Geometric Series A geometric sequence is a list of numbers with a constant ratio between consecutive terms. In other words, to get the next term in a geometric sequence, multiply the previous term by some constant number. Example of a geometric sequence: 1, 2, 4, 8, 16, 32, 64 The first term in the sequence is 1. Multiply 1 by 2 to get the second term, 2. Multiply the second term by 2 to get the third term, 4. Etc. Each term in the sequence is 2 times the previous term. This constant multiplier, 2, is called the ratio. If we were to add up all the terms in a geometric sequence, we get a geometric series. Using our previous geometric sequence example, the geometric series would be: Written in sigma notation: k 1 k=1 We will get more practice with sigma notation later, but just keep in mind that it s super useful for cutting down the amount of stuff we have to write in order to describe a series. In fact, sometimes our series will be infinite, meaning we couldn t even write the whole series out if we wanted to or not!

9 For now, it s far more important to be able to calculate a partial sum of a geometric series. Whether our series is infinite or finite, we can sum an arbitrary number of terms in the series to calculate a partial sum. If, for example, we add the first 5 terms of a series, we are calculating the 5 th partial sum of the series. The formula for calculating a partial sum of a geometric series is as follows: Formula for Partial Sum of a Geometric Series S n = a 1(1 r n ), where r 1 1 r Here is what the symbols in the formula mean: S n : the sum of the first n terms of the series, called the n th partial sum. (for example, the sum of the first 3 terms in the series would be written S 3 ). a 1 : the first term in the series. r: the ratio. (the constant number that each term is multiplied by to get the next term in the series) r n : the ratio, r, raised to the n th power. (this is the same n in the subscript for S n ) Find the 4 th partial sum of the series Use the formula to find the sum of a geometric series for the first 4 terms, given: a 1 = 1 r = 2 n = 4 (since we are finding the sum of the first 4 terms) So, S 4 = (1(1 24 )) 1 2 = 15 1 = 15 Thus, the 4 th partial sum of the geometric series is 15.

10 Let s try another example (notice that the language of the question changes, but we are being asked to find the same thing). Find the sum of the first 27 terms of the series Use our geometric series formula, given: a 1 = 1 2 r = 1 2 n = 27 Plugging it all in, S 27 = 1 2 (1 (1 2 )27 ) Thus, the sum of the first 27 terms of the geometric series is approximately

11 Practice: calculating partial sums of geometric series 1. Find the 27 th partial sum of the series Add the first 15 terms of the series