Find the square roots of 9. 3 and – 3 POSITIVE VALUE

Simplify We need to be careful…these are all Principle Square Roots. We must guarantee positive answers for all of them. Can you guarantee (x – 2) is positive? Now it is positive! Factor! Can’t take the square root unless there is 1 term. ABSOLUTE VALUE BAR RULE If an even root simplifies to a variable with an odd power, then we need absolute values around the answer. “GAZINTA” RULE Divide the root number into the power on the inside. 2 GAZINTA 8, 4 times. No absolute value bars needed. 2 GAZINTA 6, 3 times. Need absolute value bars. Odd power. |

“GAZINTA” RULE “3 GAZINTA 3” 1 time. “3 GAZINTA 6” 2 times. “3 GAZINTA 9” 3 times. “3 GAZINTA 15” 5 times.

Not Possible to have 4 factors that are the same equal a negative! “GAZINTA” RULE

Be careful!

The denominator is the root The numerator is the power

Can’t simplify because there are 2 terms inside the radical. Simplify first then multiply. The roots must be the same in order to multiply the radicands together.

2 GAZINTA RULE 2 Remember the Absolute value bar rule. Even root simplifies a variable expression to an odd power!

GAZINTA RULE We may want to think of a way to clean this up! GAZINTA RULE Time to clean! Since 432 is an even number we can agree that 2 is a factor, but will there be four factors of 2. Try 432 divided by 2 4. We saw that 27 breaks down to three 3’s. We need 4, so don’t break it down. Absolute value bar rule.

FACTORGAZINTA RULE Since 800 is an even number we can agree that 2 is a factor, but will there be five factors of 2. Try 800 divided by 2 5. GAZINTA RULE Since 256 is an even number we can agree that 2 is a factor, but will there be six factors of 2. Try 256 divided by 2 6. Remember the Absolute value bar rule. Even root simplifies a variable expression to an odd power! Absolute value bar rule.

Multiply inside with inside, outside with outside! Write as one radical and simplify. Absolute value bar rule.

Simplify first then divide. All variables will represent positive values in this section!

Radicals are not allowed in the denominator because they are IRRATIONAL numbers that have decimals that never end. The PHILOSOPHY: “JUST ENOUGH TO GET BY” finally comes true! Cube root is under the MATH button SIMPLIFY FIRST!!

Break down the denominator into prime factorization. Multiply by just enough 2’s, 3’s, x’s and y’s to get groups of 3.

in simplified form! Variables must be the same.Radicands must be the same. Powers on the variables must be the same. Roots on the radicals must be the same.

No like radicals.

Make a Note! (a + b)(a – b) = a 2 – b 2

Always use ( )’s around two or more terms! Multiply top and bottom by the conjugate of the bottom. Multiply bottoms 1 st. From the last example… (a + b)(a – b) = a 2 – b 2 Bottom doesn’t factor and nothing can cancel. Done! 5 – 3 = 2. FOIL the top and CLT…looking for a GCF of 2 on the Top.

Fractional exponents will be very helpful! Convert. There is a pattern for a short cut! 1. Multiply the roots for a common root. 2. Multiply the 1 st power to the 2 nd root as an exponent. 3. Multiply the 2 nd power to the 1 st root and add to the exponent. We need LCD to add fractional exponents.

Fractional exponents will be very helpful! Convert. There is a pattern for a short cut! 1. Multiply the roots for a common root. 2. Multiply the Top power to the bottom root minus Bottom power times top root as the exponent. 24 and 2 can be reduce by 2, their GCF.

FALSE TRUE! ( ) 3 3 WHOA!!! Wait a minute. Principle Square Roots are always positive! NO SOLUTION

NEGATIVEEVEN Even root radical are always equal to a positive value. If it is a negative, then no solution = POSITIVE…KEEP GOING + 3 = + 3

Solve for x. – 7 = – 7 3 NEGATIVE! NO SOLUTION – 6 = – 6 Cube Roots can be Negative Keep Going! + 5 = + 5 2

Solve for x. – 5 = – 5 NEGATIVE? I don’t know! But I know that x > 5 to make it positive! F.O.I.L. Set = 0 Combine Like Terms – x – 7 = – x – 7 FACTOR

Solve for x. The radical on the left is ISOLATED so square both sides…REMEMBER to FOIL the right side! The right side promises to be positive if x > 3. Combine Like Terms and ISOLATE the radical. – x + 2 = – x + 2 The radical term is isolated. Don’t divide by 2 it will create fractions. Square both sides again. REMEMBER to FOIL the left side and distribute the 4. Combine Like Terms and set equal to zero. Factor and solve for x. – 4x + 12 = – 4x + 12

Solve for x. Graph and check. Set the equation equal to zero. To the calculator. Find the Y= button and press it. Type in the above equation without the = 0. Graph in the Standard window by hitting ZOOM 6. The graph is very close to the x-axis. Hit the TRACE button to see the x and y coordinates at the bottom of the screen. In the upper left hand corner will be the equation of the graph and the coordinates are at the bottom. Notice that there is no y value. Remember we said that x > to 3. We won’t see a y until we get to 3. Type in a 3 and hit ENTER. Y = 0 means 3 is the right answer. Type in a 7 and hit ENTER. Y = 0 means 3 is the right answer.

Solve for x. The radical on the left is ISOLATED so square both sides…REMEMBER to FOIL the right side! The right side promises to be positive if x > 7. Combine Like Terms and ISOLATE the radical. The radical term is isolated. Divide by 4 because it divides evenly. Square both sides again. – x + 3 = – x = + 7 Since our answer is 11, it will not fit in the WINDOW. Hit the WINDOW button and change the Xmax to 15 and hit GRAPH. TRACE and type in 11 to check it.

a b c leg hypotenuse Remember…positive answers! a = b = c Exact ValueDecimal Value

b b b b Draw in a line to make a right triangle. a = b b c Solve for c. Label the sides with our triple ratio. Just found out what b is equal to! Solve for b.

Draw in a line to make a right triangle. 2a2a2a2a 2a2a aa b c = 2a a b Solve for b. Label the sides with our triple ratio. Just found out what a is equal to! Solve for a.

The distance formula can be derived from the Pythagorean Theorem. Build a right triangle. c a b = d There is no difference on which coordinates are subtracted first. What is important that when you square the difference it will be positive. The Midpoint Formula is going to be considered as the “Average between the two points.”

Find the distance and the midpoint between (-3, 5) and (9, -11). If M(3, 7) is the midpoint of segment AB, find the location of B when A is located at (-6, 10). Set the Midpoint Formula = (3, 7) and substitute in the coordinates of A. Multiply both equations by = + 6 – 10 = – 10 Short-cut Subtract the given endpoint * 2

Given. Notice a pattern! i to an odd power is always equal to i, just need to determine the sign. i to an even power is always equal to 1, just need to determine the sign. Time to determine the sign. Odd number of negatives multiplied together is negative and an Even number of negatives is a positive. Find the number of i 2 are in i n. EVEN Whole number = + ODD Whole number = –

Simplify radicals with negatives.

Real Number imaginary Number Definition of Complex Numbers. Complex Numbers. Real Numbers.Imaginary Numbers. When the directions read, “Leave the answers in a + bi form.” The answer will have to include a zero if there is no real number or imaginary number. For example. If the answer is 2, then we write the answer as 2 + 0i. If the answer is -5i, then we write the answer as 0 – 5i.

Combine Like Terms. Treat i like a variable. Distribute the minus sign.

Complex Conjugate Product Rule. Rationalize the Denominator above rule! The denominator is a single term, just multiply by i top and bottom.