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Chapter P Prerequisites: Fundamental Concepts of Algebra

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P.1 Algebraic Expressions & Real Numbers

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Objectives Evaluate algebraic expressions Use mathematical models Find the intersection of 2 sets Find the union of 2 sets Recognize subsets of the real numbers Use Inequality symbols Evaluate absolute value Use absolute value to express distance Identify properties of the real numbers Simplify algebraic expressions

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Intersection of Sets What they have in common A = {all tall children} B = {all girls} A intersect B = {all tall girls} All children that are girls AND are tall

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Union of Sets Combination of everything in both sets A = {all tall children} B = {all girls} A union B = {all girls OR tall children} = {all girls and all tall boys}

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A group of biology majors are taking Biology I & Chem. I. A group of chemistry majors are taking Calculus, Chem. I and Physics I. The Physics majors enrolled in Calculus, Physics I, and Chem I. What is the intersection of the 3 groups? 1.Students in biology, chemistry, & physics. 2.Students in chemistry. 3.Students in calculus. 4.Students in physics.

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Absolute Value x represents the distance between x and zero Distance is always a positive quantity, therefore going left or right x units results in a distance of x units x - 2 represents the distance between x and 2 Distance is again always positive. (i.e. the distance between 2 and 3 is 1 and the distance between 2 and 1 is 1) = = 1

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Real numbers are a field Commutative (addition & multiplication) Associative (addition & multiplication) Identity (additive = 0 & multiplicative = 1) Inverse (additive = -x & multiplicative = 1/x) Distributive (multiplication over addition) ALL these properties are useful when manipulating algebraic expressions & equations

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P.2 Exponents and Scientific Notation

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Objectives Use the product and quotient rules Use the zero-exponential rule Use the negative-exponent rule Use the power rule Find the power of a product Find the power of a quotient Simplify exponential expressions Use scientific notation

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Product & Quotient Rules

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Example Simplify:

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Quotient Rule explains zero-exponent rule Any real number divided by itself (except 0) equals 1 If x is any nonzero number & y is an exponent:

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Quotient Rule Explains Negative Exponent Rule

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Working with Negative Exponents In general, expressions are not considered simplified when negative exponents are present. A negative exponent in the denominator becomes positive when moved to the numerator A negative exponent in the numerator becomes positive when moved to the denominator

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Raising an Exponent to an Exponent (Power Rule) Exponents are multiplied WHY?

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When is your expression simplified? No negative exponents are present Each base appears only once No parentheses remain Example:

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Scientific Notation What is it? A number greater than or equal to 1 & less than 10 (either pos. or neg.) multiplied by 10 raised to an exponent Example:

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Why Use Scientific Notation? It allows us to express very large numbers or very small numbers in a more concise manner. Diminishes the error in writing very large or small numbers by eliminating the need to have all zeros written. (easy to have one too many or too few zeros)

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Converting into Scientific Notation

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Rules of Thumb Count decimal places you move to place the decimal to the right of one non-zero digit Large numbers are represented by multiplying by ten raised to a positive exponent Small numbers are represented by multiplying by ten raised to a negative exponent

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P.3 Radicals & Rational Exponents

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Objectives Evaluate square roots Simplify (nth root of nth power) Use product & quotient rules to simplify square roots Add & subtract square roots Rationalize denominators Evaluate & perform operations with higher roots Understand & use rational exponents

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Principal Square Root It is true that 4 squared and (-4) squared both equal 16, BUT the principal square root of 16 is 4 NOT -4 By convention, the radical symbol represents the positive (or PRINCIPAL) square roots of a number, thus for real numbers, x, greater than or equal to 0:

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Examples

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Multiplying & Dividing with Radicals (Roots) A product or quotient under a radical can be written as the product or quotient of separate radicals Products or quotients involving square roots can be expressed as a single square root involving products or quotients under the radical

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Adding & Subtracting Square Roots ONLY when youre taking the square root of the same number can you add or subtract square roots

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Simplify Expressions, then Add/Subtract (if possible)

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What is a conjugate? Pairs of expressions that involve the sum & the difference of two terms The conjugate of a+b is a-b Why are we interested in conjugates? When working with terms that involve square roots, the radicals are eliminated when multiplying conjugates

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Multiplying conjugates which involve square roots

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Expressions with radicals in the denominator are NOT simplified Eliminate the radical from the denominator by multiplying by the numerator and the denominator by the conjugate of the denominator Sometimes the result may not LOOK simpler!

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Other Roots The nth root of a number means what number could you raise to the nth power to get your original number? You can take an odd root of a negative number or a positive number. You can only take an even root of a positive number.

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Rules for other roots Add and subtract only same roots of same number (i.e. you can add cube roots of 3 but NOT cube roots of 3 and cube roots of 4) Multiply & divide same roots following same rules as square roots

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Subtract, if possible, & simplify:

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Expressing roots as rational exponents Any root can be expressed as a rational exponent, then rules of exponents apply

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Expressions may involve exponents AND roots If possible, its often easier to take the root first (the rational exponent), then raise the value to the other exponent

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P.4 Polynomials

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Objectives Understand the vocabulary of polynomials Add & Subtract polynomials Multiply polynomials Use FOIL in polynomial multiplication Use special products in polynomial multiplication Perform operations with polynomials in several variables

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A polynomial in x is many terms added or subtracted with each term involving a constant and x raised to a power. Only same powers of x can be added/subtracted When multiplying polynomials, the distributive property holds. (i.e. every term in one polynomial must be multiplied by every term in the other polynomial.

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Special Products

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P.5 Factoring Polynomials

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Objectives Factor out the greatest common factor Factor by grouping Factor trinomials Factor difference of squares Factor perfect square trinomials Factor sum & difference of cubes Use a general strategy for factoring Factor expressions containing fractional & negative exponents

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Factoring strategies FIRST: Look for greatest common factor Group terms (if 4 or more) to find common terms between groups If only 3 terms, rewrite into 4 terms by multiplying leading coefficient by the constant term (a times c), then rewrite bx as the sum of 2 terms whose product of their coefficients is ac (then group as in previous item)

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EXAMPLE

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Factor by Recognition Difference of Squares Difference or Sum of Cubes Opposite signs cause all middle terms to cancel out

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Factor Completely

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P.6 Rational Expressions

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Objectives Specify domain of a rational expression Simplify rational expressions Multiply rational expressions Divide rational expressions Add & subtract rational expressions Simplify complex rational expressions Simplify fractional expressions that occur in calculus Rationalize numerators

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Domain restrictions No values can be substituted in for x that would create a zero denominator or a negative value under a positive root

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Simplify rational expressions Factor numerator and denominator to cancel common terms Do NOT forget that the terms cancelled still were in the original expression, therefore must be considered when stating the domain

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Adding & Subtracting Rational Expressions Expressions MUST have a common denominator to be added/subtracted Remember when creating a common denominator, both the numerator & denominator must be multiplied by the same term, otherwise the resulting expression will NOT be equivalent to the original

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Simplify

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Rational expressions that occur in calculus To simplify this expression, you may have to rationalize the numerator (if f(x) involves a root)

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P.7 Equations Objectives Solve linear equations in one variable Solve linear equations containing fractions Solve rational equations with variables in the denominators Solve a formula for a variable Solve equations involving absolute value

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Objectives continued Solve quadratic equations by: a) Factoring b) Using the square root property c) Completing the square d) Using the quadratic formula (WHEN TO USE WHICH METHOD?) Use discriminant to determine # & type of solutions Solve application problems involving quadratics Solve radical equations

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What is a linear equation in one variable and what is solving it? Only one variable (x or y, generally) is in the equation and it is NOT squared or raised to a power other than 1. To solve the equation means to find the value (or values) that would make the equation true.

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How do we solve an equation? Eliminate parentheses (distribute!) Collect like terms (additive identity) Isolate the variable (multiplicative identity) Remember: its an EQUATION to start with, meaning the left equals the right. It will no longer be equal, if something is done to one side and not the other! CHECK your solution in the original equation: does it make it true?

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EXAMPLE 4(2x-3) = 2(x+3) 1)Distribute to eliminate parentheses 8x-12 = 2x + 6 2)Collect xs on one side & constants on the other (use additive identity) 8x(-2x) – 12(+ 12) = 2x(-2x) + 6(+ 12) 6x = 18 3)Isolate the x (use multiplicative identity) 4) Check your solution in the original 4(2(3)-3) = 2(3+3) 4(3) = 2(6) YES!!

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Rational Equations Equations that involve fractions! The variable (x) could be in the numerator of the denominator. IF the x is found in the denominator, we must consider values x canNOT take on. (i.e. zero denominator) EVEN after youve simplified an equation to eliminate the fractions, you havent eliminated the original restriction that may have been present. With fractions, EITHER eliminate the fraction OR get a common denominator (if denominators are EQUAL, so are numerators)

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Solve by getting a common denominator

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Types of Equations Conditional: True under certain conditions (could be one or several solutions) Inconsistent: Inconsistencies between the 2 sides (never true – NO solutions) Identity: One side of the equation is identical to the other (doesnt matter what x is, infinitely many solutions)

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Example Solve 3x – 6 = 3(x – 2) Notice, after distributing on the right, 3(x – 2) = 3x – 6 The left side is identical to the right. No matter what values you plug in for x, it will always be true. The solution set is: {all reals} THIS IS AN IDENTITY.

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Example Solve: 4x – 8 = 4(x – 5) Distribute on right = 4x – 20 Think: Can 4 times a number minus 8 possibly equal 4 times the same number minus 20??? NO!! If you continue to solve, you get: 0x = -12 (Can 0 times a number ever equal -12? NO! INCONSISTENCIES!! Solution: { }

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Quadratic equations Various methods to solve different quadratics

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What is a quadratic equation?

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Zero-Product Rule If the product of two or more numbers is zero, at least one of the numbers must equal zero! If AB=0, then A=0 and/or B=0 –One or both of the terms must equal zero Why is this important? It allows us an easy way to solve an equation, but FIRST make certain the expression is a product that equals zero.

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A product involves FACTORS (2x-3)(x+2)=0 2x – 3 is a factor of the expression, as is 2+x Set each factor = 0 2x – 3 = 0, thus x = 3/2 x + 2 = 0, thus x = -2 SO, if EITHER x = 3/2 or x = -2, the original expression = 0 SO, solve by FACTORING if equation, once equal to 0, is FACTORABLE

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Often, you must get expression into factored form FIRST:

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Solving with square root property When would you use this approach? –When one side of the equation is a perfect square EXAMPLE:

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Solve by Completing the Square When can you use this method? ALWAYS –However, if the expression is factorable or is already a perfect square, those methods may be more desirable HOW does it work? If you dont have a perfect square, you create one by adding a well-chosen zero (adding the same term to both sides) Decide what to add by determining what additional term would create a perfect square

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EXAMPLE

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Completing the square generalized to any quadratic equation results in the quadratic formula. When can you use it? ALWAYS. (However, it still may be easier to factor & use zero-factor property or take the square root if its already a perfect square.)

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Solve: 1.x = -1, 3 2.x = 1, -3 3.x = 2,3 4.x = 2

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What is the discriminant and why is it useful to us? The discriminant is the part of the quadratic equation that is under the radical. Based on what is under that radical, we can determine if our solution will be an integer (is whats under there a perfect square?), an irrational (is whats under there a positive number that is NOT a perfect square), or complex (is whats under there a negative number?)

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What if your equation involves a variable under a radical? In order to eliminate an nth root, you must raise both sides of the equation to the nth power. Be CERTAIN that you isolate the radical (have it on one side of the equation by itself) before you raise both sides to the nth power.

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What if the variable is found under a radical twice in an equation? Isolate one radical and raise both sides to the nth power. Then, isolate the other radical (it will not have disappeared from the other side), and raise both sides to the nth power again.

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What is x is raised to an exponent that is NOT an integer? If the variable (or expression involving a variable) is raised to the (m/n) exponent, you must isolate that expression and then raise BOTH sides to the (n/m) power. WHY?? When you raise one exponent to another, you multiply the 2 exponents.

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What if the equation involves an expression inside absolute value brackets? Recall what absolute value means: What is within those brackets could be positive or negative and still have the same overall value.

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Solve: 1.No solution. 2.{7/3} 3.{10/3, 4/3} 4.{-7/3, 7/3}

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P.8 Modeling with Equations

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Objectives Use equations to solve problems

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Solving Word Problems 1) Carefully read the problem 2) Determine what do you know and what do you want to know 3) Identify variables 4) Develop equation relating what you know & what you want to know 5) Solve the equation & check (correct?) 6) Make certain you answered the question you were being asked!

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EXAMPLE You need to drive from Chicago to your cousins house in Omaha (a distance of 550 miles) at an average 65 mph on the Interstate highway. What time should you leave if you have to be at your cousins at 3:30 pm?

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What do you want to know? –How long will it take you to drive? (x = time) –What time must you leave? What do you know? –Total distance youll travel (550 miles) –Speed (65 miles per hour) What is the relationship between known & unknown? –Distance = Rate x Time –550 miles = 65 mph x (X) (cont. on next slide)

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Did you answer the question? NO – WHEN should you leave? In order to arrive at 3:30pm, you leave 8.5 hrs earlier, which would be at 8:00 am.

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EXAMPLE You have been asked to make an aluminum can (cylindrical shape) to hold 300 ml of your product. The can is to be 10 cm high. How much aluminum (in square cm) do you need?

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What do you know? –Can holds 300 ml (the volume!) –Height = 10 cm What do you want to know? -How much material you will need (surface area). What relates the known & unknown? For cylinders:

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(example continued)

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Now find surface area! (answer the question!) (remember, a cylinder is just 2 circles and a rectangle)

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Ava purchased a new ski jacket, on sale for $ The coat had been advertised as 30% off! What was the original cost? 1.$95 2.$ $90 4.$96.50

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P.9 Linear Inequalities and Absolute Value Inequalities

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Objectives Use interval notation. Find intersections & unions of intervals. Solve linear inequalities. Recognize inequalities with no solution or all numbers as solutions. Solve compound inequalities. Solve absolute value inequalities.

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Linear inequalities For equalities, you are finding specific values that will make your expression EQUAL something. For inequalities, you are looking for values that will make your expression LESS THAN (or equal to), or MORE THAN (or equal to) something. In general, your solution set will involve an interval of values that will make the equation true, not just specific points.

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What if you have more than one inequality? If two inequalities are joined by the word AND, you are looking for values that will make BOTH true at the same time. (the INTERSECTION of the 2 sets) If two inequalities are joined by the word OR, you are looking for values that will make one inequality OR the other true (not necessarily both), therefore it is the UNION of the 2 sets.

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What IS an absolute value inequality? Recall that absolute value refers to the expression inside the brackets being either positive or negative, therefore the absolute value inequality involves 2 separate inequalities IF absolute value expression is LESS THAN a value, youre looking for values that are WITHIN that distance (intersection of the 2 inequalities) IF absolute value expression is MORE THAN a value, youre looking for values that are getting further away in both directions (union of the 2 inequalities)

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If the absolute value is greater than a number, youre considering getting further away in both directions, therefore an OR. (get further away left OR right) See next slide for example:

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If, however, the absolute value was LESS than a number (think of this as a distance problem), youre getting closer to your value and staying WITHIN a certain range. Therefore, this is an intersection problem (AND) Same problem as before, but solved as a LESS than inequality. (next slide)

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Dont leave common sense at the door! Remember to use logic! Can an absolute value ever be less than or equal to a negative value?? NO! (therefore if such an inequality were presented, the solution would be the empty set) Can an absolute value ever be more than or equal to a negative value?? YES! ALWAYS! (therefore if such an inequality were given, the solution would be all reals)

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