# 6.4E: Exercises

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## Practice Makes Perfect

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

##### Exercise 1

$$(w+4)^2$$

##### Exercise 2

$$(q+12)^2$$

$$q^2+24q+144$$

##### Exercise 3

$$(y+14)^2$$

##### Exercise 4

$$(x+\frac{2}{3})^2$$

$$x^2+\frac{4}{3}x+\frac{4}{9}$$

##### Exercise 5

$$(b−7)^2$$

##### Exercise 6

$$(y−6)^2$$

$$y^2−12y+36$$

##### Exercise 7

$$(m−15)^2$$

##### Exercise 8

$$(p−13)^2$$

$$p^2−26p+169$$

##### Exercise 9

$$(3d+1)^2$$

##### Exercise 10

$$(4a+10)^2$$

$$16a^2+80a+100$$

##### Exercise 11

$$(2q+13)^2$$

##### Exercise 12

$$(3z+15)^2$$

$$9z^2+65z+125$$

##### Exercise 13

$$(3x−y)^2$$

##### Exercise 14

$$(2y−3z)^2$$

$$4y^2−12yz+9z^2$$

##### Exercise 15

$$(15x−17y)^2$$

##### Exercise 16

$$(18x−19y)^2$$

$$164x^2−136xy+181y^2$$

##### Exercise 17

$$(3x2+2)^2$$

##### Exercise 18

$$(5u^2+9)^2$$

$$25u^4+90u^2+81$$

##### Exercise 19

$$(4y^3−2)^2$$

##### Exercise 20

$$(8p^3−3)^2$$

$$64p^6−48p^3+9$$

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

##### Exercise 21

$$(m−7)(m+7)$$

##### Exercise 22

$$(c−5)(c+5)$$

$$c^2−25$$

##### Exercise 23

$$(x+34)(x−34)$$

##### Exercise 24

$$(b+\frac{6}{7})(b−\frac{6}{7})$$

$$b^2−\frac{36}{49}$$

##### Exercise 25

$$(5k+6)(5k−6)$$

##### Exercise 26

$$(8j+4)(8j−4)$$

$$64j^2−16$$

##### Exercise 27

$$(11k+4)(11k−4)$$

##### Exercise 28

$$(9c+5)(9c−5)$$

$$81c^2−25$$

##### Exercise 29

$$(11−b)(11+b)$$

##### Exercise 30

$$(13−q)(13+q)$$

$$169−q^2$$

##### Exercise 31

$$(5−3x)(5+3x)$$

##### Exercise 32

$$(4−6y)(4+6y)$$

$$16−36y^2$$

##### Exercise 33

$$(9c−2d)(9c+2d)$$

##### Exercise 34

$$(7w+10x)(7w−10x)$$

$$49w^2−100x^2$$

##### Exercise 35

$$(m+\frac{2}{3}n)(m−\frac{2}{3}n)$$

##### Exercise 36

$$(p+\frac{4}{5}q)(p−\frac{4}{5}q)$$

$$p^2−\frac{16}{25}q^2$$

##### Exercise 37

$$(ab−4)(ab+4)$$

##### Exercise 38

$$(xy−9)(xy+9)$$

$$x^{2}y^2−81$$

##### Exercise 39

$$(uv−\frac{3}{5})(uv+\frac{3}{5})$$

##### Exercise 40

$$(rs−\frac{2}{7})(rs+\frac{2}{7})$$

$$r^{2}s^2−\frac{4}{49}$$

##### Exercise 41

$$(2x^2−3y^4)(2x^2+3y^4)$$

##### Exercise 42

$$(6m^3−4n^5)(6m^3+4n^5)$$

$$36m^6−16n^{10}$$

##### Exercise 43

$$(12p^3−11q^2)(12p^3+11q^2)$$

##### Exercise 44

$$(15m^2−8n^4)(15m^2+8n^4)$$

$$225m^4−64n^8$$

​​​​​​Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

##### Exercise 45

a. $$(p−3)(p+3)$$

b. $$(t−9)^2$$

c. $$(m+n)^2$$

d. $$(2x+y)(x−2y)$$

##### Exercise 46

a. $$(2r+12)^2$$

b. $$(3p+8)(3p−8)$$

c. $$(7a+b)(a−7b)$$

d. $$(k−6)^2$$

a. $$4r^2+48r+144$$

b. $$9p^2−64$$

c. $$7a^2−48ab−7b^2$$

d. $$k^2−12k+36$$

##### Exercise 47

a. $$(a^5−7b)^2$$

b. $$(x^2+8y)(8x−y^2)$$

c. $$(r^6+s^6)(r^6−s^6)$$

d. $$(y^4+2z)^2$$

##### Exercise 48

a. $$(x^5+y^5)(x^5−y^5)$$

b. $$(m^3−8n)^2$$

c. $$(9p+8q)^2$$

d. $$(r^2−s^3)(r^3+s^2)$$

a. $$x^{10}−y^{10}$$

b. $$m^6−16m^{3}n+64n^2$$

c. $$81p^2+144pq+64q^2$$

d. $$r^5+r^{2}s^2−r^{3}s^3−s^5$$

## Everyday Math

##### Exercise 49

Mental math You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as 50−3 and 53 as 50+3

1. Multiply (50−3)(50+3) by using the product of conjugates pattern, $$(a−b)(a+b)=a^2−b^2$$
2. Multiply 47·53 without using a calculator.
3. Which way is easier for you? Why?
##### Exercise 50

Mental math You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as 60+5.

1. Multiply $$(60+5)^2$$ by using the binomial squares pattern, $$(a+b)^2=a^2+2ab+b^2$$
2. Square 65 without using a calculator.
3. Which way is easier for you? Why?
1. 4,225
2. 4,225

## Writing Exercises

##### Exercise 51

How do you decide which pattern to use?

##### Exercise 52

Why does $$(a+b)^2$$ result in a trinomial, but (a−b)(a+b) result in a binomial?

##### Exercise 53

Marta did the following work on her homework paper:

$\begin{array}{c} {(3−y)^2}\\ {3^2−y^2}\\ {9−y^2}\\ \nonumber \end{array}$

Explain what is wrong with Marta’s work.

##### Exercise 54

Use the order of operations to show that $$(3+5)^2$$ is 64, and then use that numerical example to explain why $$(a+b)^2 \ne a^2+b^2$$