Math / Algebra

Question11. $\qquad$ $(a+3 b)(a+3 b)=$
12. $\qquad$ $(3 x y+6)(3 x y-6)=$
13. $\qquad$ $(m-2)(m-2)=$
14. $\qquad$ $(3 t-2)(3 t-2)=$
15. $\qquad$ $(x-5)(x+5)=$
16. $\qquad$ $\left(8 r^{2} s^{2}-4\right)\left(8 r^{2} s^{2}-4\right)=$
17. $\qquad$ $(a+b)^{2}=$
18. $\qquad$ $(a+b)(a-b)=$
19. $\qquad$ $(2 a b-c)^{2}=$
20. $\qquad$ $(2 a b+c)(2 a b-c)=$

Studdy Solution

STEP 1

1. We are given multiple algebraic expressions to simplify or expand.
2. We will use algebraic identities such as the square of a binomial and the difference of squares.
3. Each expression will be treated independently.

STEP 2

1. Simplify or expand each expression using appropriate algebraic identities.
2. Calculate or express the result for each expression.

STEP 3

Expression 11: $(a+3b)(a+3b)$
This is a square of a binomial, which can be expanded using the identity $(x+y)^2 = x^2 + 2xy + y^2$ .
$(a+3b)^2 = a^2 + 2(a)(3b) + (3b)^2$
$= a^2 + 6ab + 9b^2$

STEP 4

Expression 12: $(3xy+6)(3xy-6)$
This is a difference of squares, which can be simplified using the identity $(x+y)(x-y) = x^2 - y^2$ .
$(3xy+6)(3xy-6) = (3xy)^2 - 6^2$
$= 9x^2y^2 - 36$

STEP 5

Expression 13: $(m-2)(m-2)$
This is a square of a binomial, which can be expanded using the identity $(x-y)^2 = x^2 - 2xy + y^2$ .
$(m-2)^2 = m^2 - 2(m)(2) + 2^2$
$= m^2 - 4m + 4$

STEP 6

Expression 14: $(3t-2)(3t-2)$
This is a square of a binomial, which can be expanded using the identity $(x-y)^2 = x^2 - 2xy + y^2$ .
$(3t-2)^2 = (3t)^2 - 2(3t)(2) + 2^2$
$= 9t^2 - 12t + 4$

STEP 7

Expression 15: $(x-5)(x+5)$
This is a difference of squares, which can be simplified using the identity $(x-y)(x+y) = x^2 - y^2$ .
$(x-5)(x+5) = x^2 - 5^2$
$= x^2 - 25$

STEP 8

Expression 16: $(8r^2s^2-4)(8r^2s^2-4)$
This is a square of a binomial, which can be expanded using the identity $(x-y)^2 = x^2 - 2xy + y^2$ .
$(8r^2s^2-4)^2 = (8r^2s^2)^2 - 2(8r^2s^2)(4) + 4^2$
$= 64r^4s^4 - 64r^2s^2 + 16$

STEP 9

Expression 17: $(a+b)^2$
This is a square of a binomial, which can be expanded using the identity $(x+y)^2 = x^2 + 2xy + y^2$ .
$(a+b)^2 = a^2 + 2ab + b^2$

STEP 10

Expression 18: $(a+b)(a-b)$
This is a difference of squares, which can be simplified using the identity $(x+y)(x-y) = x^2 - y^2$ .
$(a+b)(a-b) = a^2 - b^2$

STEP 11

Expression 19: $(2ab-c)^2$
This is a square of a binomial, which can be expanded using the identity $(x-y)^2 = x^2 - 2xy + y^2$ .
$(2ab-c)^2 = (2ab)^2 - 2(2ab)(c) + c^2$
$= 4a^2b^2 - 4abc + c^2$

STEP 12

Expression 20: $(2ab+c)(2ab-c)$
This is a difference of squares, which can be simplified using the identity $(x+y)(x-y) = x^2 - y^2$ .
$(2ab+c)(2ab-c) = (2ab)^2 - c^2$
$= 4a^2b^2 - c^2$

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Studdy Solution

STEP 1

1. We are given multiple algebraic expressions to simplify or expand.
2. We will use algebraic identities such as the square of a binomial and the difference of squares.
3. Each expression will be treated independently.

STEP 2

1. Simplify or expand each expression using appropriate algebraic identities.
2. Calculate or express the result for each expression.

STEP 3

Expression 11: $(a+3b)(a+3b)$
This is a square of a binomial, which can be expanded using the identity $(x+y)^2 = x^2 + 2xy + y^2$ .
$(a+3b)^2 = a^2 + 2(a)(3b) + (3b)^2$
$= a^2 + 6ab + 9b^2$

STEP 4

Expression 12: $(3xy+6)(3xy-6)$
This is a difference of squares, which can be simplified using the identity $(x+y)(x-y) = x^2 - y^2$ .
$(3xy+6)(3xy-6) = (3xy)^2 - 6^2$
$= 9x^2y^2 - 36$

STEP 5

Expression 13: $(m-2)(m-2)$
This is a square of a binomial, which can be expanded using the identity $(x-y)^2 = x^2 - 2xy + y^2$ .
$(m-2)^2 = m^2 - 2(m)(2) + 2^2$
$= m^2 - 4m + 4$

STEP 6

Expression 14: $(3t-2)(3t-2)$
This is a square of a binomial, which can be expanded using the identity $(x-y)^2 = x^2 - 2xy + y^2$ .
$(3t-2)^2 = (3t)^2 - 2(3t)(2) + 2^2$
$= 9t^2 - 12t + 4$

STEP 7

Expression 15: $(x-5)(x+5)$
This is a difference of squares, which can be simplified using the identity $(x-y)(x+y) = x^2 - y^2$ .
$(x-5)(x+5) = x^2 - 5^2$
$= x^2 - 25$

STEP 8

Expression 16: $(8r^2s^2-4)(8r^2s^2-4)$
This is a square of a binomial, which can be expanded using the identity $(x-y)^2 = x^2 - 2xy + y^2$ .
$(8r^2s^2-4)^2 = (8r^2s^2)^2 - 2(8r^2s^2)(4) + 4^2$
$= 64r^4s^4 - 64r^2s^2 + 16$

STEP 9

Expression 17: $(a+b)^2$
This is a square of a binomial, which can be expanded using the identity $(x+y)^2 = x^2 + 2xy + y^2$ .
$(a+b)^2 = a^2 + 2ab + b^2$

STEP 10

Expression 18: $(a+b)(a-b)$
This is a difference of squares, which can be simplified using the identity $(x+y)(x-y) = x^2 - y^2$ .
$(a+b)(a-b) = a^2 - b^2$

STEP 11

Expression 19: $(2ab-c)^2$
This is a square of a binomial, which can be expanded using the identity $(x-y)^2 = x^2 - 2xy + y^2$ .
$(2ab-c)^2 = (2ab)^2 - 2(2ab)(c) + c^2$
$= 4a^2b^2 - 4abc + c^2$

STEP 12

Expression 20: $(2ab+c)(2ab-c)$
This is a difference of squares, which can be simplified using the identity $(x+y)(x-y) = x^2 - y^2$ .
$(2ab+c)(2ab-c) = (2ab)^2 - c^2$
$= 4a^2b^2 - c^2$

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Studdy Solution

STEP 1

1. We are given multiple algebraic expressions to simplify or expand.2. We will use algebraic identities such as the square of a binomial and the difference of squares.3. Each expression will be treated independently.

STEP 2

1. Simplify or expand each expression using appropriate algebraic identities.2. Calculate or express the result for each expression.

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

Expression 15: (x−5)(x+5)(x-5)(x+5)(x−5)(x+5)This is a difference of squares, which can be simplified using the identity (x−y)(x+y)=x2−y2(x-y)(x+y) = x^2 - y^2(x−y)(x+y)=x2−y2.(x−5)(x+5)=x2−52 (x-5)(x+5) = x^2 - 5^2 (x−5)(x+5)=x2−52=x2−25 = x^2 - 25 =x2−25

STEP 8

STEP 9

Expression 17: (a+b)2(a+b)^2(a+b)2This is a square of a binomial, which can be expanded using the identity (x+y)2=x2+2xy+y2(x+y)^2 = x^2 + 2xy + y^2(x+y)2=x2+2xy+y2.(a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 (a+b)2=a2+2ab+b2

STEP 10

Expression 18: (a+b)(a−b)(a+b)(a-b)(a+b)(a−b)This is a difference of squares, which can be simplified using the identity (x+y)(x−y)=x2−y2(x+y)(x-y) = x^2 - y^2(x+y)(x−y)=x2−y2.(a+b)(a−b)=a2−b2 (a+b)(a-b) = a^2 - b^2 (a+b)(a−b)=a2−b2

STEP 11

STEP 12

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We are given multiple algebraic expressions to simplify or expand.
2. We will use algebraic identities such as the square of a binomial and the difference of squares.
3. Each expression will be treated independently.

1. Simplify or expand each expression using appropriate algebraic identities.
2. Calculate or express the result for each expression.

Expression 15: $(x-5)(x+5)$
This is a difference of squares, which can be simplified using the identity $(x-y)(x+y) = x^2 - y^2$ .
$(x-5)(x+5) = x^2 - 5^2$
$= x^2 - 25$

Expression 17: $(a+b)^2$
This is a square of a binomial, which can be expanded using the identity $(x+y)^2 = x^2 + 2xy + y^2$ .
$(a+b)^2 = a^2 + 2ab + b^2$

Expression 18: $(a+b)(a-b)$
This is a difference of squares, which can be simplified using the identity $(x+y)(x-y) = x^2 - y^2$ .
$(a+b)(a-b) = a^2 - b^2$