simplify 2200(1+(0.03)/(12))^{72}

{ "query": { "display": "simplify $$2200\\left(1+\\frac{0.03}{12}\\right)^{72}$$", "symbolab_question": "SIMPLIFY#simplify 2200(1+\\frac{0.03}{12})^{72}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "\\frac{11\\cdot 401^{72}}{2^{285}\\cdot 25^{71}}", "decimal": "2633.28662…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2200\\left(1+\\frac{0.03}{12}\\right)^{72}=\\frac{11\\cdot\\:401^{72}}{2^{285}\\cdot\\:25^{71}}$$", "input": "2200\\left(1+\\frac{0.03}{12}\\right)^{72}", "steps": [ { "type": "interim", "title": "Simplify $$1+\\frac{0.03}{12}:{\\quad}\\frac{401}{400}$$", "input": "1+\\frac{0.03}{12}", "steps": [ { "type": "interim", "title": "$$\\frac{0.03}{12}=\\frac{3}{1200}$$", "input": "\\frac{0.03}{12}", "steps": [ { "type": "step", "primary": "Multiply and divide by 10 for every number after the decimal point.<br/>There are $$2$$ digits to the right of the decimal point, therefore multiply and divide by $$100$$", "result": "=\\frac{100\\cdot\\:0.03}{100\\cdot\\:12}" }, { "type": "step", "primary": "Multiply the numbers: $$100\\cdot\\:0.03=3$$", "result": "=\\frac{3}{100\\cdot\\:12}" }, { "type": "step", "primary": "Multiply the numbers: $$100\\cdot\\:12=1200$$", "result": "=\\frac{3}{1200}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BvsXV7Z6LhRm027gU63z91nyYRz18HvB+rp63mPitc/9ovYKijQYhJDCbxu/nAOJVS/N3lkz0U9Z+Xo4w1qOqaN6Hv6MoTMtvtU0IQwXdn8fbPcfpRdR9gYR1ocHxfkFTxNjE5RWpzDYCq2vcN7miCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=1+\\frac{3}{1200}" }, { "type": "interim", "title": "$$1+\\frac{3}{1200}=\\frac{401}{400}$$", "input": "1+\\frac{3}{1200}", "steps": [ { "type": "interim", "title": "Cancel $$\\frac{3}{1200}:{\\quad}\\frac{1}{400}$$", "input": "\\frac{3}{1200}", "steps": [ { "type": "step", "primary": "Cancel the numbers: $$\\frac{3}{1200}=\\frac{1}{400}$$", "result": "=\\frac{1}{400}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmH9pVF6zgLOizNu9rEA+gsDnzlbPZjyKgy1eUCFsLd5+LDIf9i9cO5d68tWTbtF9pK5smlJv/KMZt2xj10a6w+2+DWDoJJmVGC26S6WFWg67Lm3rcQq6QXlNVWYFIJDYF9an45VhVADtB9Or9y5Puw=" } }, { "type": "step", "result": "=1+\\frac{1}{400}" }, { "type": "interim", "title": "Convert $$1\\:$$to fraction $$:{\\quad}\\frac{400}{400}$$", "input": "1", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$a=\\frac{ab}{b}$$", "result": "=\\frac{1\\cdot\\:400}{400}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:400=400$$", "result": "=\\frac{400}{400}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Join Convert Element to Fractions Title 1Eq" } }, { "type": "step", "result": "=\\frac{400}{400}+\\frac{1}{400}" }, { "type": "interim", "title": "$$\\frac{400}{400}+\\frac{1}{400}=\\frac{401}{400}$$", "input": "\\frac{400}{400}+\\frac{1}{400}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}$$", "result": "=\\frac{400+1}{400}" }, { "type": "step", "primary": "Add the numbers: $$400+1=401$$", "result": "=\\frac{401}{400}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z7beuy2PQRHP30f8Gxb6OK21ZQmfWm9kYyulovS6kAV8kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcauhx5fQSxi0V5oVxp/FBV0yKjeh7+jKEzLb7VNCEMF3Z/A+rWRdvSj7/YJDLoSYKTNMe//gZ/+6HDoOS+y4HXhbWsK/2g3ilTVeYbvd7k28UzJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=\\frac{401}{400}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7O69JzAGDTAHFH3pFMNP2X3aNvcWr77e522ghWA7MrpejkVi15I8rBefLi4Iyt2wrtbwsiwlGw5ydTVktbDuZJT/L0MoYg+CUn6oyL3EO7YoSdmijC5i1TlYSC90WCrYKh3b61oFpJ+SbW3gNfPzfPF9an45VhVADtB9Or9y5Puw=" } }, { "type": "step", "result": "=\\frac{401}{400}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DHBL4Z40a9XE8b/dXN+jlosugzgTwrRKoyiMlIzXDHjMwViaLUXkeD+JukROhWdjcdSL2ppDBzAObYGuHtxpN/8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUlxGXARNX9/39ddoRRm4hhKUhoTYePExu56LfvWkvxh0w=" } }, { "type": "step", "result": "=2200\\left(\\frac{401}{400}\\right)^{72}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "secondary": [ "$$\\left(\\frac{401}{400}\\right)^{72}=\\frac{401^{72}}{400^{72}}$$" ], "result": "=2200\\cdot\\:\\frac{401^{72}}{400^{72}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$2200\\cdot\\:\\frac{401^{72}}{400^{72}}=\\frac{11\\cdot\\:401^{72}}{2^{285}\\cdot\\:25^{71}}$$", "input": "2200\\cdot\\:\\frac{401^{72}}{400^{72}}", "steps": [ { "type": "interim", "title": "Convert $$2200\\:$$to fraction $$:{\\quad}\\frac{2200}{1}$$", "input": "2200", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2200=\\frac{2200}{1}$$", "result": "=\\frac{2200}{1}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Join Convert Element to Fractions Title 1Eq" } }, { "type": "step", "result": "=\\frac{2200}{1}\\cdot\\:\\frac{401^{72}}{400^{72}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{b}\\cdot\\:\\frac{c}{d}=\\frac{a\\:\\cdot\\:\\:c}{b\\:\\cdot\\:\\:d}$$", "result": "=\\frac{2200\\cdot\\:401^{72}}{1\\cdot\\:400^{72}}" }, { "type": "interim", "title": "Cancel $$\\frac{2200\\cdot\\:401^{72}}{1\\cdot\\:400^{72}}:{\\quad}\\frac{11\\cdot\\:401^{72}}{2^{285}\\cdot\\:25^{71}}$$", "input": "\\frac{2200\\cdot\\:401^{72}}{1\\cdot\\:400^{72}}", "steps": [ { "type": "step", "primary": "Apply rule: $$1\\cdot\\:a=a$$", "secondary": [ "$$1\\cdot\\:400^{72}=400^{72}$$" ], "result": "=\\frac{2200\\cdot\\:401^{72}}{400^{72}}" }, { "type": "step", "primary": "Factor the number: $$2200=200\\cdot\\:11$$", "result": "=\\frac{200\\cdot\\:11\\cdot\\:401^{72}}{400^{72}}" }, { "type": "interim", "title": "Factor $$400^{72}:{\\quad}200^{72}\\cdot\\:2^{72}$$", "input": "400^{72}", "steps": [ { "type": "step", "primary": "Factor the number: $$400=200\\cdot\\:2$$", "result": "=\\left(200\\cdot\\:2\\right)^{72}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(200\\cdot\\:2\\right)^{72}=200^{72}\\cdot\\:2^{72}$$" ], "result": "=200^{72}\\cdot\\:2^{72}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "=\\frac{200\\cdot\\:11\\cdot\\:401^{72}}{200^{72}\\cdot\\:2^{72}}" }, { "type": "interim", "title": "$$\\frac{200}{200^{72}}=\\frac{1}{200^{71}}$$", "input": "\\frac{200}{200^{72}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "result": "=\\frac{1}{200^{72-1}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$72-1=71$$", "result": "=\\frac{1}{200^{71}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72UXN5IztRGdG1b29QlOzZiuWSAjrpOUoXH4YLwNIgDQJQJZuTAY5js+oqjdT8kslyjkc6gFusKCp4/vOrhbnKOIjTogT1Z+wVCKspyLrZquA3+3MYjGE0qOauqoQp45tNMYTwfIkQjhrTkX3/AqgQw+zwuk/nU0fYXBcir8orFawiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{11\\cdot\\:401^{72}}{200^{71}\\cdot\\:2^{72}}" }, { "type": "interim", "title": "$$200^{71}\\cdot\\:2^{72}=2^{285}\\cdot\\:25^{71}$$", "input": "200^{71}\\cdot\\:2^{72}", "steps": [ { "type": "step", "primary": "Factor integer $$200=2^{3}\\cdot\\:25$$", "result": "=\\left(2^{3}\\cdot\\:25\\right)^{71}\\cdot\\:2^{72}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(ab\\right)^{c}=a^{c}b^{c}$$", "secondary": [ "$$\\left(2^{3}\\cdot\\:25\\right)^{71}=\\left(2^{3}\\right)^{71}\\cdot\\:25^{71}$$" ], "result": "=\\left(2^{3}\\right)^{71}\\cdot\\:25^{71}\\cdot\\:2^{72}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "secondary": [ "$$\\left(2^{3}\\right)^{71}=2^{3\\cdot\\:71}$$" ], "result": "=2^{3\\cdot\\:71}\\cdot\\:25^{71}\\cdot\\:2^{72}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2^{213}\\cdot\\:25^{71}\\cdot\\:2^{72}" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$2^{213}\\cdot\\:2^{72}=\\:2^{213+72}$$" ], "result": "=25^{71}\\cdot\\:2^{213+72}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$213+72=285$$", "result": "=25^{71}\\cdot\\:2^{285}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vToAkj/BKreOy3oD+s/7/4rMD93PjQUR7oT2FeWEs60JQJZuTAY5js+oqjdT8kslcNGMY1NtxVpU+o52UVuzOcIRqx7aFtoqEwPqzGVruc656qsftAwKFdqyMpQYmP2gA7R3lzqpYx56/fCTtftE1RLV7FIukGJp5QmYROn7X6BMzKXh8Xzk6UFhZlN4oJqi" } }, { "type": "step", "result": "=\\frac{11\\cdot\\:401^{72}}{2^{285}\\cdot\\:25^{71}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Cancel Title 1Eq" } }, { "type": "step", "result": "=\\frac{11\\cdot\\:401^{72}}{2^{285}\\cdot\\:25^{71}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qB8Ug+5fRzswzayg5WpsPOVj3/YVQ2LqRiRWW+VA9rorlkgI66TlKFx+GC8DSIA0CUCWbkwGOY7PqKo3U/JLJazWwd2OgDf3pOoIqJQ1wuHRnh3rliO7dupcT3QpWyfbAk5+MmkmQVUpcF4h73UGdyluC+t5wk6f37N/Brqo2lszuAAm9UUBmnMmimENvgGaH3ApBdSeumNxDFeP909Wd1HGpVLovJql+zp9pgbbSJObMA2VR5hjAkYPFkroX5QH6DWuoekI+vDDvId/6cfFe7CXZvfTltaOvU01VHVTDmc=" } }, { "type": "step", "result": "=\\frac{11\\cdot\\:401^{72}}{2^{285}\\cdot\\:25^{71}}" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }