simplify 4/(2-i)

{ "query": { "display": "simplify $$\\frac{4}{2-i}$$", "symbolab_question": "SIMPLIFY#simplify \\frac{4}{2-i}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "\\frac{8}{5}+\\frac{4}{5}i", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Simplify $$\\frac{4}{2-i}:{\\quad}\\frac{8}{5}+\\frac{4}{5}i$$", "input": "\\frac{4}{2-i}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{2+i}{2+i}$$", "result": "=\\frac{4\\left(2+i\\right)}{\\left(2-i\\right)\\left(2+i\\right)}" }, { "type": "interim", "title": "Simplify $$\\frac{4\\left(2+i\\right)}{\\left(2-i\\right)\\left(2+i\\right)}:{\\quad}\\frac{8+4i}{5}$$", "input": "\\frac{4\\left(2+i\\right)}{\\left(2-i\\right)\\left(2+i\\right)}", "steps": [ { "type": "interim", "title": "$$4\\left(2+i\\right)=8+4i$$", "input": "4\\left(2+i\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\left(2+i\\right):{\\quad}8+4i$$", "input": "4\\left(2+i\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$4\\left(2+i\\right)=4\\cdot\\:2+4i$$" ], "result": "=4\\cdot\\:2+4i", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8+4i" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gZVI6MH2Qp2jM5ewxcDAhgsBmUOGgB0VAXoAXik9qC9EoApuujwRq/Ms3wa4gNT89ZVSO0B13oSpK+zJfqv7kB7MWKURN+43KCOzRc+RXaG8mxbsU5O8r6yl4Zp13ZtM" } }, { "type": "step", "result": "=8+4i" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\left(2-i\\right)\\left(2+i\\right)=5$$", "input": "\\left(2-i\\right)\\left(2+i\\right)", "steps": [ { "type": "step", "primary": "Apply complex arithmetic rule: $$\\left(a+bi\\right)\\left(a-bi\\right)=a^{2}+b^{2}$$", "result": "=2^{2}+\\left(-1\\right)^{2}" }, { "type": "interim", "title": "$$2^{2}+\\left(-1\\right)^{2}=5$$", "input": "2^{2}+\\left(-1\\right)^{2}", "steps": [ { "type": "interim", "title": "$$2^{2}=4$$", "input": "2^{2}", "steps": [ { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:\\quad$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "result": "=1" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=4+1" }, { "type": "step", "primary": "Add the numbers: $$4+1=5$$", "result": "=5" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s749OqtYD6C31VvcpsKWZOsgCWKUbvV6WK3fDUgFtg3Q88wjMomSjDqu25eW2gKyOu/COFlUvA93NcQfHx1F5YKgc7h0fqvZ5cbGiIcqgrYZUkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=5" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{8+4i}{5}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "step", "result": "=\\frac{8+4i}{5}" }, { "type": "interim", "title": "Rewrite in standard complex form: $$\\frac{8}{5}+\\frac{4}{5}i$$", "input": "\\frac{8+4i}{5}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a+bi}{c}=\\frac{a}{c}+\\frac{b}{c}i$$", "result": "=\\frac{8}{5}+\\frac{4}{5}i" } ], "meta": { "solvingClass": "Solver2", "interimType": "Rewrite In Complex Form Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYlOKRXpcvIBpvnFjdDGqJ/LRvzTB6Xm4ZezI9lY2wEfTa4GmGEyX8B4Y1y12qJzXmaN6Hv6MoTMtvtU0IQwXdn+1KB/rFi0F8EgDe2LWm5KD5M3oocynl/c4EI6wi7JIsw5OmnH/HDImMywVwBG91Io=" } }, { "type": "step", "result": "=\\frac{8}{5}+\\frac{4}{5}i" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }