• – HERE ARE SOME MORE EXAMPLES OF USING THE DISTRIBUTIVE PROPERTY TO SIMPLIFY EXPRESSIONS. IN THE FIRST EXAMPLE WE HAVE -9P x THE QUANTITY P + 2. SO, WE WANT TO DISTRIBUTE -9P, WHICH MEANS WE WANT TO MULTIPLY -9P AND P AS WELL AS -9P AND 2. SO, WE’D HAVE -9P x P + -9 x 2. WELL, -9P x P IS GOING TO BE -9P TO THE SECOND OR -9P SQUARED, AND THEN WE’LL HAVE + (-18), BUT INSTEAD OF LEAVING IT IN THIS FORM AS + AND -, IT’S MORE COMMON TO WRITE THIS AS SUBTRACTING A POSITIVE. SO, THIS IS EQUIVALENT TO -9P SQUARED – (+18).…

• – WELCOME TO TWO EXAMPLES INVOLVING THE ADDITION PROPERTY OF DEFINITE INTEGRALS. TO HELP ILLUSTRATE THESE EXAMPLES WE’LL ASSUME F OF X IS THIS FUNCTION HERE, GRAPHED IN RED, WHICH IS A NONNEGATIVE FUNCTION. WE’RE GIVEN THE DEFINITE INTEGRAL OF F OF X FROM 1 TO 6=THE DEFINITE INTEGRAL OF F OF X FROM 1 TO 4 + THE DEFINITE INTEGRAL OF F OF X FROM “A” TO B, AND WE’RE ASKED TO FIND “A” AND B. BECAUSE F OF X IS NONNEGATIVE, IF WE INTEGRATE F OF X FROM 1 TO 6 IT WOULD GIVE US THE AREA OF THIS BLUE SHADED REGION, WHICH MEANS THE SUM OF THESE…

• – WE WANT TO USE THE DISTRIBUTIVE PROPERTY TO SIMPLIFY THE GIVEN EXPRESSIONS. SO, HERE WE HAVE 5 x THE QUANTITY X + 2. SO, WE NEED TO MULTIPLY THE 5 AND THE X AS WELL AS THE 5 AND THE 2, AND SOMETIMES YOU’LL HEAR DISTRIBUTION REFERRED TO AS MULTIPLICATION ACROSS ADDITION OR SUBTRACTION, AND THIS IS WHY. SO, WE’LL HAVE 5 x X + 5 x 2. WELL, 5 x X WOULD BE 5X, AND 5 x 2=10. SO, WE HAVE 5X + 10. IN THE SECOND EXAMPLE WE’RE DISTRIBUTING +3. SO WE’LL HAVE +3 x X – +3 x 9. SO WE’LL HAVE 3 x X –…

• We want to simplify the following expressions completely. The first expression is two x raised to the fourth power. Let’s first simplify this by expanding, and then we’ll simplify it again using the power property of exponents. Notice we have an exponent of four, which means we have four factors of the base, which is two x. So this is equal to two x, times two x, times two x, times two x. Once expanded, notice how we can see we have four factors of two and four factors of x, so we can write this as two to the fourth times x to the fourth, and we can evaluate…