simplify pi/6-pi/4

{ "query": { "display": "simplify $$\\frac{π}{6}-\\frac{π}{4}$$", "symbolab_question": "SIMPLIFY#simplify \\frac{π}{6}-\\frac{π}{4}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "-\\frac{π}{12}", "decimal": "-0.26179…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{π}{6}-\\frac{π}{4}=-\\frac{π}{12}$$", "input": "\\frac{π}{6}-\\frac{π}{4}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$6,\\:4:{\\quad}12$$", "input": "6,\\:4", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$6:{\\quad}2\\cdot\\:3$$", "input": "6", "steps": [ { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRuUHkFwKrCGUG/pR2kioRow/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1AjXz67i9oO9i25G22wINi" } }, { "type": "interim", "title": "Prime factorization of $$4:{\\quad}2\\cdot\\:2$$", "input": "4", "steps": [ { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp39fF/zAtU5baHQ1hwgXA+n" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$6$$ or $$4$$", "result": "=2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2\\cdot\\:3=12$$", "result": "=12" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$12$$" }, { "type": "step", "primary": "For $$\\frac{π}{6}:\\:$$multiply the denominator and numerator by $$2$$", "result": "\\frac{π}{6}=\\frac{π2}{6\\cdot\\:2}=\\frac{π2}{12}" }, { "type": "step", "primary": "For $$\\frac{π}{4}:\\:$$multiply the denominator and numerator by $$3$$", "result": "\\frac{π}{4}=\\frac{π3}{4\\cdot\\:3}=\\frac{π3}{12}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{π2}{12}-\\frac{π3}{12}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{π2-π3}{12}" }, { "type": "step", "primary": "Add similar elements: $$2π-3π=-π$$", "result": "=\\frac{-π}{12}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{π}{12}" } ], "meta": { "solvingClass": "Solver" } }, "meta": { "showVerify": true } }