Algebra and Geometry Review – Exercise 02

QUESTION 1: introduction to the quotient rule of exponents

Simplify y^7/y^4

EXPLANATION: The exponents tell us how many y's to multiply

                y^7/y^4 = Y*y*y*y*y*y*y / y*y*y*y

                = y*y*y / 1          Canceling gives us the following

                = y^3

ANSWER: y^3

QUESTION 2: Multiplying binomials with leading coefficients of 1

Multiply and Simplify your answer: (u-2)(u+7)

EXPLANATION:

                We want to remove the parentheses from the product (u-2)(u+7)

                We first multiply each term in the first factor (u-2) by each term in the second factor (u+7) using FOIL (First, Outer, Inner, Last).

                F:            Multiply the two First terms: u*u = u^2

                O:           Multiply the two Outside terms: u*7 = u7

                I :            Multiply the two Inside terms: -2*u = -2u

                L:             Multiply the two Last terms: -2*7 = -14

               

The product (u-2)(u+7) is then equal to the sum of these terms:

                (u-2)(u+7) = u^2 + 7u + - 2u – 14

                = u^2 + 5u -14

ANSWER: u^2 + 5u -14

QUESTION 3: Product rule with positive exponents

Multiply and Simplify your answer as much as possible: 6y^3(-2y^5)

EXPLANATION:

                We must do the multiplication 6y^3 times -2y^5

                We can rewrite this product as follows:

                6y^3(-2y^5) = 6*y^3*(-2)*y^5

                = 6*(-2)*y^3*y^5

                Then, we multiply the coefficients and use the product rule to multiply the variable terms

                6*(-2)*y^3*y^5 = -12y^(3+5)

                = -12y^8

ANSWER: -12y^8

QUESTION 4: Multiplying binomials with leading coefficients greater than 1

                Multiply and Simplify your answer: (5b – 4)(3b – 7)

EXPLANATION:

We can multiply 5b – 4 by 3b – 7 using the FOIL (First, Outer, Inner, Last) method

F:            Multiply the two First terms: 5b*3b = 15b^2

O:           Multiply the two Outside terms: 5b * (-7) = -35b

I:             Multiply the two Inside terms: (-4)*3b = -12b

L:             Multiply the two Last terms: (-4)*(-7) = 28

               

The product (5b – 4)(3b – 7) is then equal to the sum of these terms

(5b – 4)(3b – 7) = 15b^2 – 35b – 12b + 28

= 15b^2 – 47b + 28

ANSWER: 15b^2 – 47b + 28

QUESTION 5: Multiplying conjugate binomials

                Multiply and Simplify your answer: (z + 9)(z – 9)

EXPLANATION:

                We are multiplying a sum (z + 9) and a difference (z – 9) of the same two terms, z and 9.

                We can use the following fact: (A + B)(A – B) = A^2 – B^2

                Here is what we get:

                (z + 9)(z – 9) = z^2 – 9^2

                = z^2 – 81

ANSWER: z^2 – 81