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Step 1.Find two factors whose product is the given number, and use these numbers to create two branches. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.If a factor is not prime, write it as the product of two factors and continue the process. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Write the composite number as the product of all the circled primes. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Find the Least Common Multiple by Listing MultiplesStep 1.List several multiples of each number.Step 2.Look for the smallest number that appears on both lists.Step 3.This number is the LCM. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.List several multiples of each number. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Look for the smallest number that appears on both lists. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.This number is the LCM. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Find the Least Common Multiple Using the Prime Factors MethodStep 1.Write each number as a product of primes.Step 2.List the primes of each number. Match primes vertically when possible.Step 3.Bring down the columns.Step 4.Multiply the factors. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Write each number as a product of primes. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.List the primes of each number. Match primes vertically when possible. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Bring down the columns. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Multiply the factors. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
NotationââââââââââââââââââââââThe result isâ¦âa+bthe sum ofaandbâaâbthe difference ofaandbâa·b,ab,(a)(b)(a)b,a(b)the product ofaandbâa÷b,a/b,ab,bathe quotient ofaandbâa+bthe sum ofaandbâaâbthe difference ofaandbâa·b,ab,(a)(b)(a)b,a(b)the product ofaandbâ... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Inequalityâa<bis readâais less thanbâais to the left ofbon the number lineâa>bis readâais greater thanbâais to the right ofbon the number lineâa<bis readâais less thanbâais to the left ofbon the number lineâa>bis readâais greater thanbâais to the right ofbon the number line | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Inequality SymbolsâââââââââââââââââWordsâaâbaisnot equal tobâa<baisless thanbâaâ¤baisless than or equal tobâa>baisgreater thanbâaâ¥baisgreater than or equal tobâaâbaisnot equal tobâa<baisless thanbâaâ¤baisless than or equal tobâa>baisgreater thanbâaâ¥baisgreat... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Grouping SymbolsParentheses()()Brackets[][]Braces{}{} | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Parentheses()() | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Brackets[][] | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Braces{}{} | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Exponential Notationananmeans multiplyaaby itself,nntimes. The expressionananis readaato thenthnthpower. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
ananmeans multiplyaaby itself,nntimes. The expressionananis readaato thenthnthpower. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Order of Operations:When simplifying mathematical expressions perform the operations in the following order:Step 1.Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.Step 2.Exponents: Simplify all expressions with... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Exponents: Simplify all expressions with exponents. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Combine Like TermsStep 1.Identify like terms.Step 2.Rearrange the expression so like terms are together.Step 3.Add or subtract the coefficients and keep the same variable for each group of like terms. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Identify like terms. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Rearrange the expression so like terms are together. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Add or subtract the coefficients and keep the same variable for each group of like terms. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Addition of Positive and Negative Integers5+3â5+(â3)8â8both positive,both negative,sum positivesum negativeâ5+35+(â3)â22different signs,different signs,more negativesmore positivessum negativesum positive5+3â5+(â3)8â8both positive,both negative,sum positivesum negativeâ5+35+(â3)â22different sign... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Property of Absolute Value:|n|â¥0|n|â¥0for all numbers. Absolute values are always greater than or equal to zero! | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Subtraction of Integers5â3â5â(â3)2â25positives5negativestake away3positivestake away3negatives2 positives2 negativesâ5â35â(â3)â885negatives, want to5positives, want tosubtract3positivessubtract3negativesneed neutral pairsneed neutral pairs5â3â5â(â3)2â25positives5negativestake away3positive... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Subtraction Property:Subtracting a number is the same as adding its opposite. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Multiplication and Division of Two Signed NumbersSame signsâProduct is positiveDifferent signsâProduct is negative | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Same signsâProduct is positive | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Different signsâProduct is negative | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Strategy for ApplicationsStep 1.Identify what you are asked to find.Step 2.Write a phrase that gives the information to find it.Step 3.Translate the phrase to an expression.Step 4.Simplify the expression.Step 5.Answer the question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Identify what you are asked to find. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Write a phrase that gives the information to find it. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Translate the phrase to an expression. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Simplify the expression. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 5.Answer the question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Equivalent Fractions Property:Ifa,b,ca,b,care numbers wherebâ0,câ0,bâ0,câ0,thenab=a·cb·cab=a·cb·canda·cb·c=ab.a·cb·c=ab. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Fraction Division:Ifa,b,candda,b,canddare numbers wherebâ0,câ0,anddâ0,bâ0,câ0,anddâ0,thenab÷cd=ab·dc.ab÷cd=ab·dc.To divide fractions, multiply the first fraction by the reciprocal of the second. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Fraction Multiplication:Ifa,b,candda,b,canddare numbers wherebâ0,anddâ0,bâ0,anddâ0,thenab·cd=acbd.ab·cd=acbd.To multiply fractions, multiply the numerators and multiply the denominators. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Placement of Negative Sign in a Fraction:For any positive numbersaandb,aandb,âab=aâb=âab.âab=aâb=âab. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Property of One:aa=1;aa=1;Any number, except zero, divided by itself is one. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Simplify a FractionStep 1.Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.Step 2.Simplify using the equivalent fractions property by dividing out common factors.Step 3.Multiply any remaining factors. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Simplify using the equivalent fractions property by dividing out common factors. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Multiply any remaining factors. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Simplify an Expression with a Fraction BarStep 1.Simplify the expression in the numerator. Simplify the expression in the denominator.Step 2.Simplify the fraction. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Simplify the expression in the numerator. Simplify the expression in the denominator. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Simplify the fraction. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Fraction Addition and Subtraction:Ifa,b,andca,b,andcare numbers wherecâ0,câ0,thenac+bc=a+bcac+bc=a+bcandacâbc=aâbc.acâbc=aâbc.To add or subtract fractions, add or subtract the numerators and place the result over the common denominator. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Strategy for Adding or Subtracting FractionsStep 1.Do they have a common denominator?Yesâgo to step 2.NoâRewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.Step 2.Add or subtract the fractions.Step 3.Simplify,... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Do they have a common denominator?Yesâgo to step 2.NoâRewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Add or subtract the fractions. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Simplify Complex FractionsStep 1.Simplify the numerator.Step 2.Simplify the denominator.Step 3.Divide the numerator by the denominator. Simplify if possible. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Simplify the numerator. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Simplify the denominator. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Divide the numerator by the denominator. Simplify if possible. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Name a DecimalStep 1.Name the number to the left of the decimal point.Step 2.Write âandâ for the decimal point.Step 3.Name the ânumberâ part to the right of the decimal point as if it were a whole number.Step 4.Name the decimal place of the last digit. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Name the number to the left of the decimal point. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Write âandâ for the decimal point. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Name the ânumberâ part to the right of the decimal point as if it were a whole number. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Name the decimal place of the last digit. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Write a DecimalStep 1.Look for the word âandââit locates the decimal point. Place a decimal point under the word âand.â Translate the words before âandâ into the whole number and place it to the left of the decimal point. If there is no âand,â write a â0â with a decimal point to its right.Step 2.M... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Look for the word âandââit locates the decimal point. Place a decimal point under the word âand.â Translate the words before âandâ into the whole number and place it to the left of the decimal point. If there is no âand,â write a â0â with a decimal point to its right. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Translate the words after âandâ into the number to the right of the decimal point. Write the number in the spacesâputting the final digit in the last place. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Fill in zeros for place holders as needed. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Round a DecimalStep 1.Locate the given place value and mark it with an arrow.Step 2.Underline the digit to the right of the place value.Step 3.Is this digit greater than or equal to 5? Yesâadd 1 to the digit in the given place value. Noâdonotchange the digit in the given place value.Step 4.Rewrite the number, delet... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Locate the given place value and mark it with an arrow. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Underline the digit to the right of the place value. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Is this digit greater than or equal to 5? Yesâadd 1 to the digit in the given place value. Noâdonotchange the digit in the given place value. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Rewrite the number, deleting all digits to the right of the rounding digit. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Add or Subtract DecimalsStep 1.Write the numbers so the decimal points line up vertically.Step 2.Use zeros as place holders, as needed.Step 3.Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Write the numbers so the decimal points line up vertically. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Use zeros as place holders, as needed. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Multiply DecimalsStep 1.Determine the sign of the product.Step 2.Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.Step 3.Place the decimal point. The number of decimal places in the product is the sum of the decimal... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Determine the sign of the product. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Place the decimal point. The number of decimal places in the product is the sum of the decimal places in the factors. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Write the product with the appropriate sign. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Multiply a Decimal by a Power of TenStep 1.Move the decimal point to the right the same number of places as the number of zeros in the power of 10.Step 2.Add zeros at the end of the number as needed. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Move the decimal point to the right the same number of places as the number of zeros in the power of 10. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Add zeros at the end of the number as needed. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Divide DecimalsStep 1.Determine the sign of the quotient.Step 2.Make the divisor a whole number by âmovingâ the decimal point all the way to the right. âMoveâ the decimal point in the dividend the same number of places - adding zeros as needed.Step 3.Divide. Place the decimal point in the quotient above the dec... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Determine the sign of the quotient. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Make the divisor a whole number by âmovingâ the decimal point all the way to the right. âMoveâ the decimal point in the dividend the same number of places - adding zeros as needed. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Divide. Place the decimal point in the quotient above the decimal point in the dividend. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Write the quotient with the appropriate sign. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Convert a Decimal to a Proper FractionStep 1.Determine the place value of the final digit.Step 2.Write the fraction: numeratorâthe ânumbersâ to the right of the decimal point; denominatorâthe place value corresponding to the final digit. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Determine the place value of the final digit. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Write the fraction: numeratorâthe ânumbersâ to the right of the decimal point; denominatorâthe place value corresponding to the final digit. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
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