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It summarizes arithmetic operations properties (commutative, associative, distributive, identity) and key fraction and equation manipulation strategies such as cross multiplication, inverting both sides, and cancelling terms. It further covers exponents and radicals, including domain constraints, exponent/log rules, and factoring formulas for quadratic, cubic, and power patterns.","Harold’s Algebra Cheat Sheet  \n19 August 2026  \nArithmetic Operations (+,–, ⦁, ÷)  \n􀝀  \n\n| Property | Equation |  |  |  |  |  |  | Tips |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| Commutative | 􀜽 + 􀜾 = 􀜾 + 􀜽\u003Cbr>􀜽 · 􀜾 = 􀜾 · 􀜽 |  |  |  |  |  |  | Reordering does not change the answer |\n| Associative | (􀜽 + 􀜾) + 􀜿 = 􀜽 + (􀜾 + 􀜿)\u003Cbr>(􀜽 · 􀜾) · 􀜿 = 􀜽 · (􀜾 · 􀜿) |  |  |  |  |  |  | Regrouping does not change the answer |\n| Distributive | 􀜽 (􀜾 ± 􀜿) = 􀜽􀜾 ± 􀜽􀜿 |  |  |  |  |  |  | Extract common terms (􀜽) |\n| Identity | 􀜽 + 0 = 􀜽\u003Cbr>􀜽 · 1 = 􀜽 |  |  |  |  |  |  | No change\u003Cbr>0 plus anything = anything\u003Cbr>1 times anything = anything |\n| Unity | (􀜽􀜽􀝊􀝊􀝕􀝕􀝐􀝐ℎℎ􀝅􀝅􀝊􀝊􀝃􀝃) = 1 |  |  |  |  |  |  | Used often Assuming 􀜽􀝊􀝕􀝐ℎ􀝅􀝊􀝃 ≠ 0 |\n| Sum (Addition) | 􀜽􀜾 ± 􀜿􀝀 = 􀜽􀝀~~ ~~􀜾􀜿 |  |  |  |  |  |  | Need a common denominator (􀜽􀜾) (􀢊􀢊) + (􀜿􀝀) (􀢈􀢈) = 􀜽􀝀~~ ~~􀜾􀜿 |\n| Difference (Subtraction) |  |  |  |  |  |  |  |  |\n| Product (Multiplication) | 􀜽 􀜿 􀜽􀜿 ~~ ~~ · ~~ ~~ = ~~ ~~\u003Cbr>􀜾 􀝀 􀜾􀝀 |  |  |  |  |  |  | Multiply numerator with numerator and denominator with denominator |\n| Quotient (Division) | 􀜽􀜾 ÷ 􀜿􀝀 = 􀜽􀜾 · 􀝀􀜿 |  |  |  |  |  |  | Invert and multiply |\n| Cross Multiplication | 􀜫􀝂 | 􀜽 􀜿 ~~ ~~ = ~~ ~~􀜾 􀝀 |  | 􀝐ℎ􀝁􀝊 􀜽􀝀 = 􀜾􀜿 |  |  |  | Used often |\n| Inverting Both Sides | 􀜫􀝂 |  | 􀜽 􀜿 ~~ ~~ = ~~ ~~􀜾 􀝀 |  | 􀝐ℎ􀝁􀝊 􀜾􀜽 = |  | 􀝀\u003Cbr>􀜿 | Same as cross multiplication, followed by division |\n| Variable × Fraction | 􀜽 (􀜾􀜿) = 􀜽􀜾􀜿 |  |  |  |  |  |  | Set 􀜽 = 􀯔1, then\u003Cbr>multiply numerator with numerator and denominator with denominator |\n| Variable ÷ Fraction | 􀜽 􀜽􀜿 ~~ ~~ = ~~ ~~(􀜾~~􀜿 ~~ ) 􀜾 |  |  |  |  |  |  | Invert (􀯕􀯖) then multiply |\n| Distribute Denominator | 􀜽 + 􀜾􀜿 |  |  |  | = 􀜽􀜿 + | 􀜾\u003Cbr>􀜿 |  | 􀜿 divides both sides of the numerator |\n| Swapping Signs | 􀜽 − 􀜾 􀜾 − 􀜽 ~~ ~~ = ~~ ~~\u003Cbr>􀜿 − 􀝀 􀝀 − 􀜿 |  |  |  |  |  |  | Multiplying both the numerator and denominator by (-1) cancels the negatives |\n| Cancelling Terms | 􀜽􀜾~~ ~~~~ ~~􀜽􀜿 = 􀜾 + 􀜿 |  |  |  |  |  |  | Assuming 􀜽 ≠ 0 |\n| Fraction over a Fraction | (􀜽~~􀜾~~)  􀜽􀝀( 􀜿 ) = 􀜾􀜿 |  |  |  |  |  |  | Invert the denominator, then multiply |\n\n\n| Fraction from Hell |  (􀜽~~􀜾~~)   (􀜿~~􀝀~~)  = 􀜽􀝀􀝂􀝃(􀝁~~􀝂~~)  􀜾􀜿􀝁ℎ(􀝃~~ℎ~~) | (((􀜰~~􀜦􀜦~~􀜰)))\u003Cbr>(((􀜦~~􀜰􀜰~~􀜦)))\u003Cbr>=\u003Cbr>􀜰􀜰􀜰􀜰􀜦􀜦􀜦􀜦 |\n| --- | --- | --- |\n\nExponents (􀢇􀢈 )  \n\n| Property | Equation |  |  | Example |\n| --- | --- | --- | --- | --- |\n| Product Rule | 􀜽􀯡 􀜽􀯠 = 􀜽􀯡+􀯠 |  |  | 􀜽 3 􀜽 2 = (􀜽 ∙ 􀜽 ∙ 􀜽)(􀜽 ∙ 􀜽) = 􀜽 3+2 = 􀜽 5 |\n| Power Rule | (􀜽􀯡 )􀯠 = 􀜽􀯡􀯠 |  |  | (􀜽3)2 = (􀜽 ∙ 􀜽 ∙ 􀜽)(􀜽 ∙ 􀜽 ∙ 􀜽) = 􀜽 3∙2 = 􀜽 6 |\n| Zero Power | 􀜽 0 = 1 |  |  | By definition, assuming 􀜽 ≠ 0 |\n| Negative Powers | 􀜽 −􀯡 1\u003Cbr>􀜽 −􀯡 | = |  1 􀜽􀯡 | 2−3 = 123 = 18 |\n|  |  | = 􀜽􀯡 |  |  |\n| Quotient Rule | 􀜽􀜽􀯡􀯠 = 􀜽􀯡−􀯠 = 􀜽1􀯠−􀯡 |  |  | 2423 = 2∙22∙22∙22 = 24−3 = 21 = 2 |\n| Product Distribution | (􀜽􀜾)􀯡 = 􀜽􀯡 􀜾 􀯡 |  |  |  |\n| Quotient Distribution | (􀜽􀜾)􀯡 = 􀜽􀜾􀯡􀯡 |  |  | (~~ ~~12)3 = 1323 = 18 |\n| Fractional Powers | 􀜽􀯡~~􀯠~~ = (􀜽1~~􀯠~~)􀯡 = (􀜽􀯡 )1~~􀯠~~ |  |  | 23~~4~~ = ( 21~~4~~)3 = (23)1~~4~~ |\n| Negative Powers | (􀜽􀜾)−􀯡 = (􀜾􀜽)􀯡 |  |  | (~~ ~~12)−3 = (~~ ~~21)3 = 8 |\n| Domain | The domain of 􀜽􀯫 is\u003Cbr>(−∞, ∞) |  |  | |\n\nRadicals ( 􀢔√􀢇)  \n\n| Property | Equation | Example |\n| --- | --- | --- |\n| Fractional Powers |    1 􀳙√􀜽 = 􀜽􀯡 | 3√2 = 21~~3~~ |\n| Power Rule (Nested Roots) | 􀳘√􀳙√􀜽 = 􀳙􀳘√􀜽 | 4√ 3√2 = 3∙4√2 = 12√2 |\n| Product Distribution | \u003Cbr>􀳙√􀜽􀜾 = 􀳙√􀜽 􀳙√􀜾 | \u003Cbr>3√14 = 3√2 3√7 |\n| Quotient Distribution | 􀳙√􀜽􀜾 = 􀜽􀜾 | 23 = 23 |\n| Odd | 􀳙√􀜽􀯡 = 􀜽 if 􀝊 is odd | Odd powers always keep their sign |\n| Even | 􀳙√􀜽􀯡 = ±􀜽 if 􀝊 is even | Even roots of even powers can be both positive and negative. Substitute to check if each solution is valid. |\n| Domain | The domain of √􀝔 is 􀝔 ≥ 0 | |\n\nLogarithms (􀜔􀜗􀜏􀢈 􀢞)  \n\n| Property | Equation |  | Example |  |\n| --- | --- | --- | --- | --- |\n| Definition | 􀢈􀢟 = 􀢞 ⟺ 􀜔􀜗􀜏 􀢈 (􀢞) = 􀢟 |  | Pulls the exponent down to level |  |\n| Example | 53 = 125 ⟺ log 5 (125) = 3 |  | Logs make large numbers very small |  |\n| Special Logs | Natural log\u003Cbr>􀝈􀝊 = Logarithme Naturel (French) |  | ln (􀝔) = log 􀯘 (􀝔)\u003Cbr>where e ","cbCaiedhZ6bG2X5f","https://ap.wps.com/l/cbCaiedhZ6bG2X5f","pdf",1536214,8,"English","en",105,"# Arithmetic Operations\n## Exponents and Exponent Rules\n## Radicals\n## Logarithms\n## Factoring Formulas","[{\"question\":\"What are the main arithmetic operation properties listed in the sheet?\",\"answer\":\"The sheet lists commutative, associative, and distributive properties, plus identity and related notes such as reordering or regrouping not changing the answer.\"},{\"question\":\"How does the sheet suggest solving equations using fractions?\",\"answer\":\"It highlights strategies like cross multiplication, inverting both sides, and cancelling terms, and includes tips such as using a common denominator when adding or subtracting fractions.\"},{\"question\":\"Which topic areas are covered beyond arithmetic and equation rules?\",\"answer\":\"The sheet covers exponents and exponent rules, radicals with domain guidance, logarithms with definitions and key rule sets, and factoring formulas including quadratic and cubic patterns.\"}]","Harold’s Algebra Cheat Sheet - Math Study Guide | PDF",1791923801,6,{"code":4,"msg":76,"data":77},"ok",{"site_id":69,"language":68,"slug":78,"title":59,"keywords":79,"description":60,"schema_data":80,"social_meta":136,"head_meta":138,"extra_data":140,"updated_unix":141},"harolds-algebra-cheat-sheet-math-study-guide","",{"@graph":81,"@context":135},[82,98,118],{"@type":83,"itemListElement":84},"BreadcrumbList",[85,89,92,95],{"item":86,"name":87,"@type":88,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":90,"name":10,"@type":88,"position":91},"https://docshare.wps.com/template/",2,{"item":93,"name":51,"@type":88,"position":94},"https://docshare.wps.com/template/general/",3,{"item":96,"name":59,"@type":88,"position":97},"https://docshare.wps.com/template/harolds-algebra-cheat-sheet-math-study-guide/571811/",4,{"url":96,"name":59,"@type":99,"image":100,"author":105,"headline":59,"publisher":107,"fileFormat":110,"inLanguage":68,"description":60,"dateModified":111,"datePublished":112,"encodingFormat":110,"isAccessibleForFree":113,"interactionStatistic":114},"DigitalDocument",{"url":101,"@type":102,"width":103,"height":104},"https://docshare.wps.com/thumbnails/harolds-algebra-cheat-sheet-math-study-guide/571811.png","ImageObject",442,249,{"name":57,"@type":106},"Person",{"url":86,"name":108,"@type":109},"DocShare","Organization","application/pdf","2026-10-25","2026-10-13",true,{"@type":115,"interactionType":116,"userInteractionCount":66},"InteractionCounter",{"@type":117},"ViewAction",{"@type":119,"mainEntity":120},"FAQPage",[121,127,131],{"name":122,"@type":123,"acceptedAnswer":124},"What are the main arithmetic operation properties listed in the sheet?","Question",{"text":125,"@type":126},"The sheet lists commutative, associative, and distributive properties, plus identity and related notes such as reordering or regrouping not changing the answer.","Answer",{"name":128,"@type":123,"acceptedAnswer":129},"How does the sheet suggest solving equations using fractions?",{"text":130,"@type":126},"It highlights strategies like cross multiplication, inverting both sides, and cancelling terms, and includes tips such as using a common denominator when adding or subtracting fractions.",{"name":132,"@type":123,"acceptedAnswer":133},"Which topic areas are covered beyond arithmetic and equation rules?",{"text":134,"@type":126},"The sheet covers exponents and exponent rules, radicals with domain guidance, logarithms with definitions and key rule sets, and factoring formulas including quadratic and cubic patterns.","https://schema.org",{"og:url":96,"og:type":137,"og:title":59,"og:site_name":108,"og:description":60},"article",{"robots":139,"canonical":96},"index,follow",{"doc_id":55,"site_id":69},1792009232]