simplify (sqrt(5)-sqrt(3))(sqrt(5)+2sqrt(3))

{ "query": { "display": "simplify $$\\left(\\sqrt{5}-\\sqrt{3}\\right)\\left(\\sqrt{5}+2\\sqrt{3}\\right)$$", "symbolab_question": "SIMPLIFY#simplify (\\sqrt{5}-\\sqrt{3})(\\sqrt{5}+2\\sqrt{3})" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "-1+\\sqrt{15}", "decimal": "2.87298…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left(\\sqrt{5}-\\sqrt{3}\\right)\\left(\\sqrt{5}+2\\sqrt{3}\\right)=-1+\\sqrt{15}$$", "input": "\\left(\\sqrt{5}-\\sqrt{3}\\right)\\left(\\sqrt{5}+2\\sqrt{3}\\right)", "steps": [ { "type": "interim", "title": "Apply FOIL method $$\\left(\\sqrt{5}-\\sqrt{3}\\right)\\left(\\sqrt{5}+2\\sqrt{3}\\right):{\\quad}-1+\\sqrt{15}$$", "input": "\\left(\\sqrt{5}-\\sqrt{3}\\right)\\left(\\sqrt{5}+2\\sqrt{3}\\right)", "steps": [ { "type": "step", "primary": "Apply FOIL method: $$\\left(a+b\\right)\\left(c+d\\right)=ac+ad+bc+bd$$", "secondary": [ "$$\\left(\\sqrt{5}-\\sqrt{3}\\right)\\left(\\sqrt{5}+2\\sqrt{3}\\right)=\\sqrt{5}\\sqrt{5}+\\sqrt{5}\\cdot\\:2\\sqrt{3}-\\sqrt{3}\\sqrt{5}-\\sqrt{3}\\cdot\\:2\\sqrt{3}$$" ], "result": "=\\sqrt{5}\\sqrt{5}+\\sqrt{5}\\cdot\\:2\\sqrt{3}-\\sqrt{3}\\sqrt{5}-\\sqrt{3}\\cdot\\:2\\sqrt{3}", "meta": { "title": { "extension": "F-First<br/>O-Outer<br/>I-Inner<br/>L-Last" }, "practiceLink": "/practice/expansion-practice#area=main&subtopic=FOIL_Basic", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\sqrt{5}\\sqrt{5}+\\sqrt{5}\\cdot\\:2\\sqrt{3}-\\sqrt{3}\\sqrt{5}-\\sqrt{3}\\cdot\\:2\\sqrt{3}=-1+\\sqrt{15}$$", "input": "\\sqrt{5}\\sqrt{5}+\\sqrt{5}\\cdot\\:2\\sqrt{3}-\\sqrt{3}\\sqrt{5}-\\sqrt{3}\\cdot\\:2\\sqrt{3}", "steps": [ { "type": "interim", "title": "$$\\sqrt{5}\\sqrt{5}=5$$", "input": "\\sqrt{5}\\sqrt{5}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{5}\\sqrt{5}=5$$" ], "result": "=5" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sqrt{5}\\cdot\\:2\\sqrt{3}=2\\sqrt{5}\\sqrt{3}$$", "input": "\\sqrt{5}\\cdot\\:2\\sqrt{3}", "steps": [ { "type": "step", "result": "=2\\sqrt{5}\\sqrt{3}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$-\\sqrt{3}\\cdot\\:2\\sqrt{3}=-6$$", "input": "-\\sqrt{3}\\cdot\\:2\\sqrt{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=-3\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$-3\\cdot\\:2=-6$$", "result": "=-6" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=5+2\\sqrt{5}\\sqrt{3}-\\sqrt{3}\\sqrt{5}-6" }, { "type": "step", "primary": "Subtract the numbers: $$5-6=-1$$", "result": "=-1+2\\sqrt{5}\\sqrt{3}-\\sqrt{3}\\sqrt{5}" }, { "type": "interim", "title": "$$-1+2\\sqrt{5}\\sqrt{3}-\\sqrt{3}\\sqrt{5}=-1+\\sqrt{15}$$", "input": "-1+2\\sqrt{5}\\sqrt{3}-\\sqrt{3}\\sqrt{5}", "steps": [ { "type": "interim", "title": "$$2\\sqrt{5}\\sqrt{3}=2\\sqrt{15}$$", "input": "2\\sqrt{5}\\sqrt{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{b}=\\sqrt{ab},\\:\\quad\\:a\\ge0,\\:b\\ge0$$", "secondary": [ "$$\\sqrt{5}\\sqrt{3}=\\sqrt{5\\cdot\\:3}$$" ], "result": "=2\\sqrt{5\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:3=15$$", "result": "=2\\sqrt{15}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sqrt{3}\\sqrt{5}=\\sqrt{15}$$", "input": "\\sqrt{3}\\sqrt{5}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{b}=\\sqrt{ab},\\:\\quad\\:a\\ge0,\\:b\\ge0$$", "secondary": [ "$$\\sqrt{3}\\sqrt{5}=\\sqrt{3\\cdot\\:5}$$" ], "result": "=\\sqrt{3\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:5=15$$", "result": "=\\sqrt{15}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=-1+2\\sqrt{15}-\\sqrt{15}" }, { "type": "step", "primary": "Add similar elements: $$2\\sqrt{15}-\\sqrt{15}=\\sqrt{15}$$", "result": "=-1+\\sqrt{15}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gtGgKh7A1QEDGLsvwPXtn6iYNjsASgR7A7ADDdo0xrmlu+1RgFMOKm5QpXsxz3R43XeO2tIUPH5Q2xrCOU6NXWtIBBIVl/9xRGvvgQHSYRBxnr1Sz5+iHyIi3WZgRGEXdLlUUcwGTBU7x2fk9SCtbgjj8R7voU/IUGOe6pTZBA3h89d4kFeDq2UyCgoi0mMpo0G4+jEUKXeQbOE1lbOh9w==" } }, { "type": "step", "result": "=-1+\\sqrt{15}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XmafnKnZ8vS4hnzkiX6j/d08QRGouW/GpnWUwndQ68Ohd3XjWtomF6O/BxtvDuIJ8qT1VSDnBlN2p5LpWu79eipFqAMDtYYPqd6fesp9MJOg+NXT3YouZe0pVU/Dj0nI3oZCq59Hq2va8/E5S/sf72YtDCKkSVExz83S81Z431ODHF/rB+QHXdRVsvySB8RZSPQX89NxX7Fh+aYyqvY8ZtcXprCavIskNPkWQmmgRYv76WA8fPhe9VV2M5NyOkLyQeEaEvI8Fa5Uax0nb1PddKNTp58YbeWMelCcqQOAfi+mxQqXKIgawW7STcfViSrVAkkHtEpkKmMewNV1c9chTLCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=-1+\\sqrt{15}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Expand Using FOIL Rule Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7O62yKKtC6+yKjUNmPFBt97Dkftmjl6ZlpFXvfRAvoW7nQ740VEMejIkIpdEtu3QdljEIcLCf4SSfCw1Z3SGzGCue+lhDRrKAl/Ka6Jx2qlkp/zrfgn506AvHp9hPf6x6pH/o5DTVlWQilOq21zRPSsRqjx9l76LKjzMNAGUYf2ajeh7+jKEzLb7VNCEMF3Z/6bMq+bZv8WUkj/92RXeHEjQJ6qB+BLL91QnaswGguzoTyU/1Qrct9SQw05k53Xv+Lzak8ryN86CRGexkc5WW9n8iMl9r2HgckMsNbv3soSA=" } }, { "type": "step", "result": "=-1+\\sqrt{15}" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }