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Elementary Algebra

Review Exercises

Elementary AlgebraReview Exercises
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Table of contents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Review Exercises

Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

588.


ⓐ 11c4−23c2+111c4−23c2+1
 ⓑ 9p3+6p2−p−59p3+6p2−p−5
 ⓒ 37x+51437x+514
 ⓓ 10
 ⓔ 2y−122y−12

589.


ⓐ a2−b2a2−b2
 ⓑ 24d324d3
 ⓒ x2+8x−10x2+8x−10
 ⓓ m2n2−2mn+6m2n2−2mn+6
 ⓔ 7y3+y2−2y−47y3+y2−2y−4

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

590.
  1. ⓐ 3x2+9x+103x2+9x+10
  2. ⓑ 14a2bc14a2bc
  3. ⓒ 6y+16y+1
  4. ⓓ n3−4n2+2n−8n3−4n2+2n−8
  5. ⓔ −19−19
591.
  1. ⓐ 5p3−8p2+10p−45p3−8p2+10p−4
  2. ⓑ −20q4−20q4
  3. ⓒ x2+6x+12x2+6x+12
  4. ⓓ 23r2s2−4rs+523r2s2−4rs+5
  5. ⓔ 100

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

592.

5y 3 + 8 y 3 5y 3 + 8 y 3

593.

−14 k + 19 k −14 k + 19 k

594.

12 q − ( −6 q ) 12 q − ( −6 q )

595.

−9 c − 18 c −9 c − 18 c

596.

12x − 4 y − 9 x 12x − 4 y − 9 x

597.

3 m 2 + 7 n 2 − 3 m 2 3 m 2 + 7 n 2 − 3 m 2

598.

6 x 2 y − 4 x + 8 x y 2 6 x 2 y − 4 x + 8 x y 2

599.

13a + b 13a + b

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

600.

( 5 x 2 + 12 x + 1 ) + ( 6 x 2 − 8 x + 3 ) ( 5 x 2 + 12 x + 1 ) + ( 6 x 2 − 8 x + 3 )

601.

( 9 p 2 − 5 p + 3 ) + ( 4 p 2 − 4 ) ( 9 p 2 − 5 p + 3 ) + ( 4 p 2 − 4 )

602.

( 10 m 2 − 8 m − 1 ) − ( 5 m 2 + m − 2 ) ( 10 m 2 − 8 m − 1 ) − ( 5 m 2 + m − 2 )

603.

( 7 y 2 − 8 y ) − ( y − 4 ) ( 7 y 2 − 8 y ) − ( y − 4 )

604.

Subtract
(3s2+10)from(15s2−2s+8)(3s2+10)from(15s2−2s+8)

605.

Find the sum of (a2+6a+9)and(5a3−7)(a2+6a+9)and(5a3−7)

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

606.

Evaluate 3y2−y+13y2−y+1 when:

  1. ⓐ y=5y=5
  2. ⓑ y=−1y=−1
  3. ⓒ y=0y=0
607.

Evaluate 10−12x10−12x when:

  1. ⓐ x=3x=3
  2. ⓑ x=0x=0
  3. ⓒ x=−1x=−1
608.

Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial −16t2+200−16t2+200 gives the height of a stone tt seconds after it is dropped from the cliff. Find the height after t=3t=3 seconds.

609.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial −4p2+460p.−4p2+460p. Find the revenue received when p=75p=75 dollars.

Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

610.

10 4 10 4

611.

17 1 17 1

612.

( 2 9 ) 2 ( 2 9 ) 2

613.

( 0.5 ) 3 ( 0.5 ) 3

614.

( −2 ) 6 ( −2 ) 6

615.

− 2 6 − 2 6

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

616.

x 4 · x 3 x 4 · x 3

617.

p 15 · p 16 p 15 · p 16

618.

4 10 · 4 6 4 10 · 4 6

619.

8 · 8 5 8 · 8 5

620.

n · n 2 · n 4 n · n 2 · n 4

621.

y c · y 3 y c · y 3

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

622.

( m 3 ) 5 ( m 3 ) 5

623.

( 5 3 ) 2 ( 5 3 ) 2

624.

( y 4 ) x ( y 4 ) x

625.

( 3 r ) s ( 3 r ) s

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

626.

( 4 a ) 2 ( 4 a ) 2

627.

( −5 y ) 3 ( −5 y ) 3

628.

( 2 m n ) 5 ( 2 m n ) 5

629.

( 10 x y z ) 3 ( 10 x y z ) 3

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

630.

( p 2 ) 5 · ( p 3 ) 6 ( p 2 ) 5 · ( p 3 ) 6

631.

( 4 a 3 b 2 ) 3 ( 4 a 3 b 2 ) 3

632.

( 5 x ) 2 ( 7 x ) ( 5 x ) 2 ( 7 x )

633.

( 2 q 3 ) 4 ( 3 q ) 2 ( 2 q 3 ) 4 ( 3 q ) 2

634.

( 1 3 x 2 ) 2 ( 1 2 x ) 3 ( 1 3 x 2 ) 2 ( 1 2 x ) 3

635.

( 2 5 m 2 n ) 3 ( 2 5 m 2 n ) 3

Multiply Monomials

In the following exercises 8, multiply the monomials.

636.

( −15 x 2 ) ( 6 x 4 ) ( −15 x 2 ) ( 6 x 4 )

637.

( −9 n 7 ) ( −16 n ) ( −9 n 7 ) ( −16 n )

638.

( 7 p 5 q 3 ) ( 8 p q 9 ) ( 7 p 5 q 3 ) ( 8 p q 9 )

639.

( 5 9 a b 2 ) ( 27 a b 3 ) ( 5 9 a b 2 ) ( 27 a b 3 )

Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

640.

7 ( a + 9 ) 7 ( a + 9 )

641.

−4 ( y + 13 ) −4 ( y + 13 )

642.

−5 ( r − 2 ) −5 ( r − 2 )

643.

p ( p + 3 ) p ( p + 3 )

644.

− m ( m + 15 ) − m ( m + 15 )

645.

−6 u ( 2 u + 7 ) −6 u ( 2 u + 7 )

646.

9 ( b 2 + 6 b + 8 ) 9 ( b 2 + 6 b + 8 )

647.

3q2(q2−7q+6)3q2(q2−7q+6) 3

648.

( 5 z − 1 ) z ( 5 z − 1 ) z

649.

( b − 4 ) · 11 ( b − 4 ) · 11

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using: ⓐ the Distributive Property, ⓑ the FOIL method, ⓒ the Vertical Method.

650.

( x − 4 ) ( x + 10 ) ( x − 4 ) ( x + 10 )

651.

( 6 y − 7 ) ( 2 y − 5 ) ( 6 y − 7 ) ( 2 y − 5 )

In the following exercises, multiply the binomials. Use any method.

652.

( x + 3 ) ( x + 9 ) ( x + 3 ) ( x + 9 )

653.

( y − 4 ) ( y − 8 ) ( y − 4 ) ( y − 8 )

654.

( p − 7 ) ( p + 4 ) ( p − 7 ) ( p + 4 )

655.

( q + 16 ) ( q − 3 ) ( q + 16 ) ( q − 3 )

656.

( 5 m − 8 ) ( 12 m + 1 ) ( 5 m − 8 ) ( 12 m + 1 )

657.

( u 2 + 6 ) ( u 2 − 5 ) ( u 2 + 6 ) ( u 2 − 5 )

658.

( 9 x − y ) ( 6 x − 5 ) ( 9 x − y ) ( 6 x − 5 )

659.

( 8 m n + 3 ) ( 2 m n − 1 ) ( 8 m n + 3 ) ( 2 m n − 1 )

Multiply a Trinomial by a Binomial

In the following exercises, multiply using ⓐ the Distributive Property, ⓑ the Vertical Method.

660.

( n + 1 ) ( n 2 + 5 n − 2 ) ( n + 1 ) ( n 2 + 5 n − 2 )

661.

( 3 x − 4 ) ( 6 x 2 + x − 10 ) ( 3 x − 4 ) ( 6 x 2 + x − 10 )

In the following exercises, multiply. Use either method.

662.

( y − 2 ) ( y 2 − 8 y + 9 ) ( y − 2 ) ( y 2 − 8 y + 9 )

663.

( 7 m + 1 ) ( m 2 − 10 m − 3 ) ( 7 m + 1 ) ( m 2 − 10 m − 3 )

Special Products

Square a Binomial Using the Binomial Squares Pattern

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

664.

( c + 11 ) 2 ( c + 11 ) 2

665.

( q − 15 ) 2 ( q − 15 ) 2

666.

( x + 1 3 ) 2 ( x + 1 3 ) 2

667.

( 8 u + 1 ) 2 ( 8 u + 1 ) 2

668.

( 3 n 3 − 2 ) 2 ( 3 n 3 − 2 ) 2

669.

( 4 a − 3 b ) 2 ( 4 a − 3 b ) 2

Multiply Conjugates Using the Product of Conjugates Pattern

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

670.

( s − 7 ) ( s + 7 ) ( s − 7 ) ( s + 7 )

671.

( y + 2 5 ) ( y − 2 5 ) ( y + 2 5 ) ( y − 2 5 )

672.

( 12 c + 13 ) ( 12 c − 13 ) ( 12 c + 13 ) ( 12 c − 13 )

673.

( 6 − r ) ( 6 + r ) ( 6 − r ) ( 6 + r )

674.

( u + 3 4 v ) ( u − 3 4 v ) ( u + 3 4 v ) ( u − 3 4 v )

675.

( 5 p 4 − 4 q 3 ) ( 5 p 4 + 4 q 3 ) ( 5 p 4 − 4 q 3 ) ( 5 p 4 + 4 q 3 )

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

676.

( 3 m + 10 ) 2 ( 3 m + 10 ) 2

677.

( 6 a + 11 ) ( 6 a − 11 ) ( 6 a + 11 ) ( 6 a − 11 )

678.

( 5 x + y ) ( x − 5 y ) ( 5 x + y ) ( x − 5 y )

679.

( c 4 + 9 d ) 2 ( c 4 + 9 d ) 2

680.

( p 5 + q 5 ) ( p 5 − q 5 ) ( p 5 + q 5 ) ( p 5 − q 5 )

681.

( a 2 + 4 b ) ( 4 a − b 2 ) ( a 2 + 4 b ) ( 4 a − b 2 )

Divide Monomials

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

682.

u 24 u 6 u 24 u 6

683.

10 25 10 5 10 25 10 5

684.

3 4 3 6 3 4 3 6

685.

v 12 v 48 v 12 v 48

686.

x x 5 x x 5

687.

5 5 8 5 5 8

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

688.

75 0 75 0

689.

x 0 x 0

690.

− 12 0 − 12 0

691.

( − 12 0 ) ( − 12 0 ) ( −12 ) 0 ( −12 ) 0

692.

25 x 0 25 x 0

693.

( 25 x ) 0 ( 25 x ) 0

694.

19 n 0 − 25 m 0 19 n 0 − 25 m 0

695.

( 19 n ) 0 − ( 25 m ) 0 ( 19 n ) 0 − ( 25 m ) 0

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

696.

( 2 5 ) 3 ( 2 5 ) 3

697.

( m 3 ) 4 ( m 3 ) 4

698.

( r s ) 8 ( r s ) 8

699.

( x 2 y ) 6 ( x 2 y ) 6

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

700.

( x 3 ) 5 x 9 ( x 3 ) 5 x 9

701.

n 10 ( n 5 ) 2 n 10 ( n 5 ) 2

702.

( q 6 q 8 ) 3 ( q 6 q 8 ) 3

703.

( r 8 r 3 ) 4 ( r 8 r 3 ) 4

704.

( c 2 d 5 ) 9 ( c 2 d 5 ) 9

705.

( 3 x 4 2 y 2 ) 5 ( 3 x 4 2 y 2 ) 5

706.

( v 3 v 9 v 6 ) 4 ( v 3 v 9 v 6 ) 4

707.

( 3 n 2 ) 4 ( −5 n 4 ) 3 ( −2 n 5 ) 2 ( 3 n 2 ) 4 ( −5 n 4 ) 3 ( −2 n 5 ) 2

Divide Monomials

In the following exercises, divide the monomials.

708.

−65 y 14 ÷ 5 y 2 −65 y 14 ÷ 5 y 2

709.

64 a 5 b 9 −16 a 10 b 3 64 a 5 b 9 −16 a 10 b 3

710.

144 x 15 y 8 z 3 18 x 10 y 2 z 12 144 x 15 y 8 z 3 18 x 10 y 2 z 12

711.

( 8 p 6 q 2 ) ( 9 p 3 q 5 ) 16 p 8 q 7 ( 8 p 6 q 2 ) ( 9 p 3 q 5 ) 16 p 8 q 7

Divide Polynomials

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

712.

42 z 2 − 18 z 6 42 z 2 − 18 z 6

713.

( 35 x 2 − 75 x ) ÷ 5 x ( 35 x 2 − 75 x ) ÷ 5 x

714.

81 n 4 + 105 n 2 −3 81 n 4 + 105 n 2 −3

715.

550 p 6 − 300 p 4 10 p 3 550 p 6 − 300 p 4 10 p 3

716.

( 63 x y 3 + 56 x 2 y 4 ) ÷ ( 7 x y ) ( 63 x y 3 + 56 x 2 y 4 ) ÷ ( 7 x y )

717.

96 a 5 b 2 − 48 a 4 b 3 − 56 a 2 b 4 8 a b 2 96 a 5 b 2 − 48 a 4 b 3 − 56 a 2 b 4 8 a b 2

718.

57 m 2 − 12 m + 1 −3 m 57 m 2 − 12 m + 1 −3 m

719.

105 y 5 + 50 y 3 − 5 y 5 y 3 105 y 5 + 50 y 3 − 5 y 5 y 3

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

720.

( k 2 − 2 k − 99 ) ÷ ( k + 9 ) ( k 2 − 2 k − 99 ) ÷ ( k + 9 )

721.

( v 2 − 16 v + 64 ) ÷ ( v − 8 ) ( v 2 − 16 v + 64 ) ÷ ( v − 8 )

722.

( 3 x 2 − 8 x − 35 ) ÷ ( x − 5 ) ( 3 x 2 − 8 x − 35 ) ÷ ( x − 5 )

723.

( n 2 − 3 n − 14 ) ÷ ( n + 3 ) ( n 2 − 3 n − 14 ) ÷ ( n + 3 )

724.

( 4 m 3 + m − 5 ) ÷ ( m − 1 ) ( 4 m 3 + m − 5 ) ÷ ( m − 1 )

725.

( u 3 − 8 ) ÷ ( u − 2 ) ( u 3 − 8 ) ÷ ( u − 2 )

Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

726.

9 −2 9 −2

727.

( −5 ) −3 ( −5 ) −3

728.

3 · 4 −3 3 · 4 −3

729.

( 6 u ) −3 ( 6 u ) −3

730.

( 2 5 ) −1 ( 2 5 ) −1

731.

( 3 4 ) −2 ( 3 4 ) −2

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

732.

p −2 · p 8 p −2 · p 8

733.

q −6 · q −5 q −6 · q −5

734.

( c −2 d ) ( c −3 d −2 ) ( c −2 d ) ( c −3 d −2 )

735.

( y 8 ) −1 ( y 8 ) −1

736.

( q −4 ) −3 ( q −4 ) −3

737.

a 8 a 12 a 8 a 12

738.

n 5 n −4 n 5 n −4

739.

r −2 r −3 r −2 r −3

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

740.

8,500,000

741.

0.00429

742.

The thickness of a dime is about 0.053 inches.

743.

In 2015, the population of the world was about 7,200,000,000 people.

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

744.

3.8 × 10 5 3.8 × 10 5

745.

1.5 × 10 10 1.5 × 10 10

746.

9.1 × 10 −7 9.1 × 10 −7

747.

5.5 × 10 −1 5.5 × 10 −1

Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

748.

( 2 × 10 5 ) ( 4 × 10 −3 ) ( 2 × 10 5 ) ( 4 × 10 −3 )

749.

( 3.5 × 10 −2 ) ( 6.2 × 10 −1 ) ( 3.5 × 10 −2 ) ( 6.2 × 10 −1 )

In the following exercises, divide and write your answer in decimal form.

750.

8 × 10 5 4 × 10 −1 8 × 10 5 4 × 10 −1

751.

9 × 10 −5 3 × 10 2 9 × 10 −5 3 × 10 2

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