Whakaoti i te 165,000=d(1-(1+.002/12))^-(30)(12)/(frac{0.08{12})}

$$165,000 = \frac { d ( 1 - ( 1 + \frac { .002 } { 12 } ) ) ^ { - ( 30 ) ( 12 ) } } { ( \frac { 0.08 } { 12 } ) }$$

Whakaoti mō d

$$d=\frac{11}{136285652597652400633072146943255896341029201131980190548843315311006893267249227122863590367756090838736813150399831954741580003320794103238355908174878733366334155884618443998783841969103173072216240445102125485099344220416740519344692695598710475488518829182397045850190391541760000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}\approx 8.071282479 \cdot 10^{-1358}$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakarohaina te \frac{0.002}{12} mā te whakarea i te taurunga me te tauraro ki te 1000.

$$165000=\frac{d\left(1-\left(1+\frac{2}{12000}\right)\right)^{-30\times 12}}{\frac{0.08}{12}}$$

Whakahekea te hautanga \frac{2}{12000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$165000=\frac{d\left(1-\left(1+\frac{1}{6000}\right)\right)^{-30\times 12}}{\frac{0.08}{12}}$$

Tāpirihia te 1 ki te \frac{1}{6000}, ka \frac{6001}{6000}.

$$165000=\frac{d\left(1-\frac{6001}{6000}\right)^{-30\times 12}}{\frac{0.08}{12}}$$

Tangohia te \frac{6001}{6000} i te 1, ka -\frac{1}{6000}.

$$165000=\frac{d\left(-\frac{1}{6000}\right)^{-30\times 12}}{\frac{0.08}{12}}$$

Whakareatia te -30 ki te 12, ka -360.

$$165000=\frac{d\left(-\frac{1}{6000}\right)^{-360}}{\frac{0.08}{12}}$$

Tātaihia te -\frac{1}{6000} mā te pū o -360, kia riro ko 13628565259765240063307214694325589634102920113198019054884331531100689326724922712286359036775609083873681315039983195474158000332079410323835590817487873336633415588461844399878384196910317307221624044510212548509934422041674051934469269559871047548851882918239704585019039154176000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$165000=\frac{d\times 13628565259765240063307214694325589634102920113198019054884331531100689326724922712286359036775609083873681315039983195474158000332079410323835590817487873336633415588461844399878384196910317307221624044510212548509934422041674051934469269559871047548851882918239704585019039154176000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{\frac{0.08}{12}}$$

Whakarohaina te \frac{0.08}{12} mā te whakarea i te taurunga me te tauraro ki te 100.

$$165000=\frac{d\times 13628565259765240063307214694325589634102920113198019054884331531100689326724922712286359036775609083873681315039983195474158000332079410323835590817487873336633415588461844399878384196910317307221624044510212548509934422041674051934469269559871047548851882918239704585019039154176000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{\frac{8}{1200}}$$

Whakahekea te hautanga \frac{8}{1200} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$165000=\frac{d\times 13628565259765240063307214694325589634102920113198019054884331531100689326724922712286359036775609083873681315039983195474158000332079410323835590817487873336633415588461844399878384196910317307221624044510212548509934422041674051934469269559871047548851882918239704585019039154176000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{\frac{1}{150}}$$

Whakawehea te d\times 13628565259765240063307214694325589634102920113198019054884331531100689326724922712286359036775609083873681315039983195474158000332079410323835590817487873336633415588461844399878384196910317307221624044510212548509934422041674051934469269559871047548851882918239704585019039154176000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 ki te \frac{1}{150}, kia riro ko d\times 2044284788964786009496082204148838445115438016979702858232649729665103399008738406842953855516341362581052197255997479321123700049811911548575338622623181000495012338269276659981757629536547596083243606676531882276490163306251107790170390433980657132327782437735955687752855873126400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$165000=d\times 2044284788964786009496082204148838445115438016979702858232649729665103399008738406842953855516341362581052197255997479321123700049811911548575338622623181000495012338269276659981757629536547596083243606676531882276490163306251107790170390433980657132327782437735955687752855873126400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$d\times 2044284788964786009496082204148838445115438016979702858232649729665103399008738406842953855516341362581052197255997479321123700049811911548575338622623181000495012338269276659981757629536547596083243606676531882276490163306251107790170390433980657132327782437735955687752855873126400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000=165000$$

Whakawehea ngā taha e rua ki te 2044284788964786009496082204148838445115438016979702858232649729665103399008738406842953855516341362581052197255997479321123700049811911548575338622623181000495012338269276659981757629536547596083243606676531882276490163306251107790170390433980657132327782437735955687752855873126400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$d=\frac{165000}{2044284788964786009496082204148838445115438016979702858232649729665103399008738406842953855516341362581052197255997479321123700049811911548575338622623181000495012338269276659981757629536547596083243606676531882276490163306251107790170390433980657132327782437735955687752855873126400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Whakahekea te hautanga \frac{165000}{2044284788964786009496082204148838445115438016979702858232649729665103399008738406842953855516341362581052197255997479321123700049811911548575338622623181000495012338269276659981757629536547596083243606676531882276490163306251107790170390433980657132327782437735955687752855873126400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 15000.

$$d=\frac{11}{136285652597652400633072146943255896341029201131980190548843315311006893267249227122863590367756090838736813150399831954741580003320794103238355908174878733366334155884618443998783841969103173072216240445102125485099344220416740519344692695598710475488518829182397045850190391541760000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$