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Quiz Questions on Exponents, Polynomials, and Radicals from SGPE Summer School 2016, Lecture notes of Algebra

A collection of quiz questions from the SGPE Summer School 2016 covering topics such as exponents, polynomials, and radicals. Students are required to simplify expressions using various rules for manipulating exponents, expand and regroup expressions, factor polynomials using integer coefficients, and perform arithmetic operations on fractions.

What you will learn

  • How do you simplify the expression x3โˆšx using the rules for manipulating exponents?
  • How do you simplify the expression x6โˆšx3 using the rules for manipulating exponents?
  • How do you write the expression (x2โˆ’2x+1)(x2+5x+6)(x2โˆ’1) in its simplest form?
  • How do you simplify the expression โˆšx4y8 using the rules for manipulating exponents?
  • How do you divide the expressions 15ch4 and 5x2yz3/3c4h using rational expressions?
  • How do you simplify the expression (x2y2x3y3)1/6?

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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Week 1 Quiz: Basics
SGPE Summer School 2016
Rules for Exponents
Question 1: Simplify the following expression using the various rules for manipulating exponents:
x6
โˆšx3
(A) โˆšx4
(B) x3
(C) โˆšx9
(D) x5.5
(E) None of the above
Answer: (C) x6
โˆšx3=x6
x
3
2=x6โˆ’
3
2=x4.5=x9
2=โˆšx9
Question 2: Simplify the following expression using the various rules for manipulating exponents:
x3โˆšx
(A) x3
2
(B) x7
(C) โˆšx4
(D) โˆšx7
(E) None of the above
Answer: (D) x3โˆšx=x3x1
2=x3+ 1
2=x7
2=โˆšx7
Question 3: Simplify the following expression using the various rules for manipulating exponents:
๎˜’x4
y8๎˜“3
(A) x7
y11
(B) x12
y24
(C) (x4โˆ’y8)3
(D) 3
qx4
y8
(E) None of the above
Answer: (B) ๎˜x4
y8๎˜‘3=x4โˆ—3
y8โˆ—3=x12
y24
1
pf3
pf4
pf5

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Download Quiz Questions on Exponents, Polynomials, and Radicals from SGPE Summer School 2016 and more Lecture notes Algebra in PDF only on Docsity!

Week 1 Quiz: Basics

SGPE Summer School 2016

Rules for Exponents

Question 1: Simplify the following expression using the various rules for manipulating exponents:

โˆš^ x^6 x^3 (A) โˆšx^4 (B) x^3 (C)

x^9 (D) x^5.^5 (E) None of the above Answer: (C) โˆšxx^63 = (^) xx 632 = x^6 โˆ’^32 = x^4.^5 = x^92 =

x^9

Question 2: Simplify the following expression using the various rules for manipulating exponents:

x^3 โˆšx

(A) x^32 (B) x^7 (C) โˆšx^4 (D)

x^7 (E) None of the above Answer: (D) x^3 โˆšx = x^3 x^12 = x3+^12 = x^72 = โˆšx^7 Question 3: Simplify the following expression using the various rules for manipulating exponents: (x 4 y^8

(A) (^) yx 117

(B) x y^1224 (C) (x^4 โˆ’ y^8 )^3

(D) 3

โˆšx 4 y^8 (E) None of the above

Answer: (B)

(x 4 y^8

= x y^48 โˆ—โˆ— 33 = x y^1224

Question 4: Simplify the following expression using the various rules for manipulating exponents:

(x^2 y^2 x^3 y^3 )^16

Answer: (x^2 y^2 x^3 y^3 )^16 = (x2+3y2+3)^16 = (x^5 y^5 )^16 = ((xy)^5 )^16 = (xy)^56 Question 5: Simplify the following expression using the various rules for manipulating exponents:

(xโˆ’ (^14) y 3 x^12

)^13

Answer:

(xโˆ’ (^14) y 3 x 12

)^13

= (xโˆ’^14 โˆ’^12 y^3 )^13 = (xโˆ’^34 y^3 )^13 = xโˆ’^14 y = (^) xy 14

Polynomials

Question 6: Expand the following expression: (2x + 7)^2 (A) 8x^2 + 18x โˆ’ 35 (B) 8x^2 + 49 (C) 4x^2 + 28x + 49 (D) 8x^2 + 18x โˆ’ 42 (E) None of the above Answer: (C) (2x + 7)^2 = (2x + 7)(2x + 7) = 4x^2 + 14x + 14x + 49 = 4x^2 + 28x + 49 Question 7: Expand and regroup the following expression: (4x + 3y)(5x^2 โˆ’ 2 xy + 6y^2 ) (A) 20x^3 + 7x^2 y + 18xy^2 + 18y^3 (B) 20x^3 โˆ’ 2 xy + 18y^3 (C) 9x^3 + 8x^2 y + 6xy^2 + 9y^3 (D) 20x^2 + 25xy + 18y^2 (E) None of the above Answer: (A) (4x + 3y)(5x^2 โˆ’ 2 xy + 6y^2 ) = 20x^3 โˆ’ 8 x^2 y + 24xy^2 + 15x^2 y โˆ’ 6 xy^2 + 18y^3 = 20x^3 + 7x^2 y + 18 xy^2 + 18y^3 Question 8: Expand and regroup the following expression:

(ax + b)^2 + (cx โˆ’ d)^2

Answer: (ax + b)^2 + (cx โˆ’ d)^2 = (ax + b)(ax + b) + (cx โˆ’ d)(cx โˆ’ d) = (a^2 x^2 + 2abx + b^2 ) + (c^2 x^2 โˆ’ 2 cdx + d^2 ) = (a^2 + c^2 )x^2 + 2(ab โˆ’ cd)x + b^2 + d^2

(E) None of the above Answer: (B) I solve these types of problems slightly differently than described in the Schaumโ€™s guide. You can decide for yourself which way you prefer. First, convert 7x^2 โˆ’ 39 x โˆ’ 18 to x^2 โˆ’ 39 x โˆ’ 7 โˆ— 18 = x^2 โˆ’ 39 x โˆ’ 126. Factor x^2 โˆ’ 39 x โˆ’ 126. Need to find c, d that satisfy cd = โˆ’126 and c + d = โˆ’39. Try -42 and 3. Now we know that x^2 โˆ’ 39 x โˆ’ 126 = (x + 3)(x โˆ’ 42). Since we multiplied by 7 in the first step, we need to divide by 7: (x +^37 )(x โˆ’ 427 )

Since 42 is divisible by 7, the second term becomes (x โˆ’ 6). However, 3 is not divisible by 7, thus we โ€œslideโ€ the 7 in front of the x. Thus the first term becomes (7x + 3). Thus 7x^2 โˆ’ 39 x โˆ’ 18 can be factored as (7x + 3)(x โˆ’ 6). N.B.: This rule works only for quadratics. Question 13: Factor the following polynomial:

16 x^2 โˆ’ 49 y^2

(A) (x + 7)(16x + 7y^2 ) (B) (4x โˆ’ 7 y^2 ) (C) (4x โˆ’ 7 y)(4x โˆ’ 7 y) (D) (4x โˆ’ 7 y)(4x + 7y) (E) None of the above Answer: (D) Answering this question requires a simply application of the โ€œdifferences-of-squareโ€ rule for factoring polynomials. Best to just memorize this formula as it generally useful. Any polynomial that is expressed as a โ€œdifference-of-squaresโ€ (i.e., anything of the form a^2 โˆ’b^2 where a and b are general expressions), factors according to (a โˆ’ b)(a + b). In this case a = โˆš 16 x^2 = 4x and b = โˆš 49 y^2 = 7y. Applying the rules yields answer D: (4x โˆ’ 7 y)(4x + 7y).

Fractions

Question 14: Multiply the following rational expressions involving quotients of binomials and reduce to lowest terms. (^) x โˆ’ 5

x + 8 ยท^

x + 2 x โˆ’ 9 (A) ((xxโˆ’+8)(5)(xx+2)โˆ’9)

(B) x x^22 โˆ’โˆ’^3 xxโˆ’โˆ’ 7210

(C) ((xxโˆ’+8)(5)(xx+3)โˆ’9)

(D) x x^22 +3โˆ’xx+72โˆ’^10 (E) None of the above Answer: (B) Work through the following algebra:

x โˆ’ 5 x + 8.

x + 2 x โˆ’ 9 =

(x โˆ’ 5)(x + 2) (x + 8)(x โˆ’ 9) =^

x^2 โˆ’ 5 x + 2x โˆ’ 10 x^2 โˆ’ 9 x + 8x โˆ’ 72 =^

x^2 โˆ’ 3 x โˆ’ 10 x^2 โˆ’ x โˆ’ 72

Question 15: Divide the following expressions:

15 ch^4 5 x^2 yz^3 รท^

3 c^4 h 55 y^2 z

(A) (^) c^553 xh (^23) zy 2

(B) (^55) c 4 hxz^2 y 22

(C) (^55) c 3 hxz^3 y 2

(D) (^) c^553 xh (^22) zy 2 (E) None of the above Answer: (A) Work through the following algebra:

15 ch^4 5 x^2 yz^3 รท^

3 c^4 h 55 y^2 z =

515 x^2 chyz^43 553 cy^42 hz^ =^

15 ch^4 5 x^2 yz^3

55 y^2 z 3 c^4 h =

5 h^3 x^2 z^2

11 y c^3 =

55 h^3 y c^3 x^2 z^2

Question 16: Add or subtract the following fractions: 12 x^2 โˆ’ 49 +^

7 x x + 7 (A) (^) x 2 12+7+xโˆ’x 42

(B) 12((xx 2 +7)+7โˆ’49)(xx+7)^3 โˆ’^49

(C) 12(x+7)+7 x (^2) โˆ’x 49 (x^2 โˆ’49)

(D) 12+7 x 2 x+49(xโˆ’7) (E) None of the above Answer: (E) Work through the following algebra:

12 x^2 โˆ’ 49 +^

7 x x + 7 =^

(x โˆ’ 7)(x + 7) +^

7 x x + 7 =

12 + 7x(x โˆ’ 7) x^2 โˆ’ 49 =

12 + 7x^2 โˆ’ 49 x) x^2 โˆ’ 49

Question 17: Write the following expression in its simplest form:

x^2 โˆ’ 2 x + 1 x + 3 ยท^

x^2 + 5x + 6 x^2 โˆ’ 1

Answer:x^2 โˆ’ x+3^2 x+1 ยท x^2 x+5 (^2) โˆ’x 1 +6 = (x xโˆ’+31) 2 ยท ((xx+3)(โˆ’1)(xx+2)+1) = (xโˆ’1)( x+1x+2)

Radicals

Question 18: Simplify the following the radicals: โˆš 81 x^8 y^6

(A) 9x^4 y^3 (B) 9โˆšx^8 y^6 (C) โˆ’ 9 x^4 y^3 (D) 9x^4 y^3 and โˆ’ 9 x^4 y^3 (E) None of the above