Ngā Tauira Pātai e Matapaki ana i te Hononga i waenga i ngā Mana me ngā Pūtake
I roto i te pāngarau, he ariā taketake ngā taupū me ngā pūtake e puta pinepine ana i roto i ngā peka pūtaiao. He ariā e pā tata ana ngā taupū me ngā pūtake, ā, e whakamahia pinepine ana hei whakahaere me te whakaoti rapanga. Ka tūhuratia e tēnei tuhinga ētahi tauira rapanga e pā ana ki te whanaungatanga i waenga i ngā taupū me ngā pūtake, me ngā kōrero taipitopito hei āwhina i te whakapakari i tō māramatanga.
Te Māramatanga Taketake mō ngā Mana me ngā Pūtake
Ko te mana he tau e puta mai ana i te whakarea i tētahi tau ki a ia anō i ngā wā e n. Hei tauira, \( a^n \) ko 'a' te pūtake, ā, ko 'n' te taupū. Hei tauira, ko te tikanga o \( 2^3 \) ko \( 2 \times 2 \times 2 = 8 \).
Ko te pūtake te mahi whakahuri o te taupū. Hei tauira, ko te pūtake tapawhā o te 9 he 3, nā te mea \(3^2 = 9 \). I te nuinga o te wā, ka tuhia he pūtake ki te āhua \(\sqrt[n]{a}\), ko 'a' te tau e pūtakehia ana, ā, ko 'n' te tohu o te pūtake.
Ngā Pātai Tauira me te Kōrero
Raru 1: Te Hononga i waenga i ngā Mana me ngā Pūtake
Pātai:
Arotakehia te uara o \( \sqrt[3]{8^3} \).
Kōrero:
Hei whakaoti i tēnei raruraru, me mārama tātou ko te mahi pūtake tapawhā (\(\sqrt[3]{ }\)) te mahi whakamuri o te whakapūtanga tapawhā (taupū 3). Me tuhi ngā mahi hei whakaoti:
1. Kia mahara ko \( 8^3 = (2^3)^3 \).
2. Mā te whakangawari, ka whiwhi tātou i te \( (2^3)^3 \).
3. I runga i te āhuatanga taupū \((a^m)^n = a^{mn}\), ka taea e tātou te whakangawari \( (2^3)^3 = 2^{3 \times 3} = 2^9 \).
4. Nō reira, ka taea te tuhi anō i te pātai kia \(\sqrt[3]{2^9}\).
Hei haere tonu, whakamahia te āhuatanga e \(\sqrt[n]{a^m} = a^{m/n}\):
5. Kātahi, \(\sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8\).
Nō reira, ko te uara o \( \sqrt[3]{8^3} = 8 \).
Pātai 2: Te Whakamahi i ngā Āhuatanga o ngā Mana me ngā Pūtake
Pātai:
Whakangāwaritia te kīanga \((\sqrt{a^4})^{3/2}\).
Kōrero:
Hei whakangawari i tēnei kīanga, ka whakamahia e tātou ngā āhuatanga o ngā taupū me ngā pūtake. Anei ngā mahi:
1. Ko te kīanga tuatahi ko \((\sqrt{a^4})^{3/2}\).
2. Kia maumahara ko te pūtake tapawhā o \( a^4 \) he ōrite ki te haurua o te mana: \(\sqrt{a^4} = (a^4)^{1/2} = a^{4 \times 1/2} = a^2\).
3. Nō reira, ka taea e tātou te tuhi anō i te kīanga hei \((a^2)^{3/2}\).
Muri iho, whakamahia te āhuatanga taupū \((a^m)^n = a^{m \times n}\):
4. \((a^2)^{3/2} = a^{2 \times 3/2} = a^3\).
Nō reira, ka whakamāmāhia te kīanga \((\sqrt{a^4})^{3/2}\) ki \(a^3\).
Raru 3: Te Huinga o ngā Tau Mana me ngā Pūtake
Pātai:
Kimihia te uara o \(\left( \sqrt{25} + \sqrt[3]{8} \right)^2\).
Kōrero:
Hei whakaoti i tēnei raruraru, me tatau motuhake ia pūtake i te tuatahi, kātahi ka tāpirihia, ā, hei whakamutunga, me tapawhā te hua:
1. Tuatahi, tatauhia te uara o \(\sqrt{25}\):
\[ \sqrt{25} = 5 \]
2. Kātahi, tatauhia te uara o \(\sqrt[3]{8}\):
\[ \sqrt[3]{8} = 2 \]
Tāpirihia ngā hua o ēnei pūtake e rua:
\[ 5 + 2 = 7 \]
Hei whakamutunga, tapawhāhia te hua:
\[ 7^2 = 49 \]
Nō reira, ko te uara o \(\left( \sqrt{25} + \sqrt[3]{8} \right)^2\) he 49.
Pātai 4: Ngā Kīanga kei roto ngā Pūtake me ngā Taupūnga Kino
Pātai:
Whakangāwaritia te kīanga \(\left( \frac{1}{\sqrt[3]{x^2}} \right)^6\).
Kōrero:
Hei whakangawari i tēnei kīanga, ka whakamahia e tātou ngā āhuatanga o ngā taupū me ngā pūtake. Tuatahi, me huri te pūtake pūtoru ki te āhua taupū:
1. Kia maumahara ko \(\sqrt[3]{x^2} = x^{2/3}\).
2. Kātahi, \(\frac{1}{\sqrt[3]{x^2}} = x^{-2/3}\).
Muri iho, whakamahia ngā āhuatanga o ngā taupū hei aromatawai i te mana o te 6 o tēnei kīanga:
3. \((x^{-2/3})^6 = x^{(-2/3) \times 6} = x^{-4}\).
Nō reira, ka whakamāmāhia te kīanga \(\left( \frac{1}{\sqrt[3]{x^2}} \right)^6\) ki \(x^{-4}\).
Raru 5: Otinga mā te whakamahi i ngā āhuatanga taketake o ngā pakiaka
Pātai:
Mena ko \(x = (\sqrt[3]{64})^{1/2}\), kimihia te uara o x.
Kōrero:
Hei whakaoti i tēnei raruraru, ka taea e tātou te whai i ēnei mahi:
1. Tātaihia te uara o \(\sqrt[3]{64}\):
\[ \sqrt[3]{64} = 4 \]
nā te mea \( 4^3 = 64 \).
2. Muri iho, tatauhia te haurua mana o te hua:
\[ (\sqrt[3]{64})^{1/2} = 4^{1/2} = \sqrt{4} = 2 \].
Nō reira, mēnā ko \( x = (\sqrt[3]{64})^{1/2} \), ko \( x = 2 \).
Whakamutunga
He pūkenga nui te mārama ki te whanaungatanga i waenga i ngā taupū me ngā pūtake i roto i te pāngarau. He maha ngā wā ka whakamahia ēnei ariā i roto i ngā momo rapanga hei whakahaere, hei aromatawai rānei i ngā kīanga. Mā te whakaharatau i ngā rapanga e pā ana ki ngā taupū me ngā pūtake, ka hohonu ake tō māramatanga me tō kaha ki te whakaoti rapanga pāngarau. Kia maumahara ki te whakaaro tonu ki ngā āhuatanga o ngā taupū me ngā pūtake i te wā e whakaoti ana i ēnei momo rapanga.
