Ngā Tauira Pātai e Matapaki ana i ngā Taupū me ngā Tauira Kōrero
Ko ngā taupū me ngā taurakitā he ariā pāngarau nui e rua e kitea pinepine ana i roto i ngā momo mara ako, pērā i te pāngarau, te pūtaiao, te ōhanga, me te hangarau. He mea nui te māramatanga pai ki ngā taupū me ngā taurakitā hei whakaoti rapanga pāngarau. Ka whakaratohia e tēnei tuhinga he tauira rapanga me ngā kōrero taipitopito e pā ana ki ngā taupū me ngā taurakitā.
Taupū
Ko te taupū he tau e whakaatu ana i te maha o ngā wā ka whakareatia he tau turanga ki a ia anō. Ko te āhua whānui o te taupū ko \(a^n\), ko \(a\) te tau matua, ā, ko \(n\) te taupū.
Tauira o ngā Raru Taupū
Pātai 1:
Whakatauhia te uara o \(2^5\).
Kōrero:
Ko te uara o \(2^5\) he 2 ina whakareatia ki a ia anō kia 5 ngā wā.
\[ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \]
Nō reira, ko te uara o \(2^5\) he 32.
Pātai 2:
Tātaihia te uara o \( (3^2) \times (3^3) \).
Kōrero:
Hei whakaoti i tēnei raruraru, ka taea e tātou te whakamahi i tētahi o ngā ture matua o ngā taupū e kī ana:
\[ a^m \times a^n = a^{m+n} \]
Nō reira,
\[ (3^2) \times (3^3) = 3^{2+3} = 3^5 = 243 \]
Nō reira, ko te uara o \( (3^2) \times (3^3) \) he 243.
Pātai 3:
Whakangāwaritia \( \frac{5^6}{5^3} \).
Kōrero:
Hei whakangawari i ngā hautau taupū me te pūtake kotahi, ka taea e tātou te whakamahi i te ture:
\[ \frac{a^m}{a^n} = a^{mn} \]
Nō reira,
\[ \frac{5^6}{5^3} = 5^{6-3} = 5^3 = 125 \]
Nō reira, ko te uara o \( \frac{5^6}{5^3} \) he 125.
Logarithma
Ko te logarithm te whakahurihanga o te taupū. I te nuinga o te wā, mēnā ko \( a^b = c \), ko \( \log_a c = b \). Arā, ko te logarithm o tētahi tau te taupū e hiahiatia ana hei tiki i taua tau mai i tētahi pūtake.
Ngā Tauira Pātai mō te Logarithm
Pātai 4:
Whakatauhia te uara o \( \log_2 32 \).
Kōrero:
Hei whakatau i te uara o \( \log_2 32 \), me kimi e tātou te uara o te taupū e whakaputa ana i te 32 ina ko te pūtake he 2.
\[ 2^5 = 32 \]
Te tikanga,
\[ \log_2 32 = 5 \]
Nō reira, ko te uara o \( \log_2 32 \) he 5.
Pātai 5:
Tātaihia te uara o \( \log_3 81 \).
Kōrero:
Hei whakatau i te uara o \( \log_3 81 \), me kimi e tātou te uara o te taupū e whakaputa ana i te 81 ina ko te pūtake he 3.
\[ 3^4 = 81 \]
Te tikanga,
\[ \log_3 81 = 4 \]
Nō reira, ko te uara o \( \log_3 81 \) he 4.
Pātai 6:
Whakangāwaritia te kīanga taupūnga \( \log(100) + \log(10) \).
Kōrero:
Ka taea e tātou te whakamahi i te ture logarithmic e kī ana:
\[ \log(a) + \log(b) = \log(ab) \]
Nō reira,
\[ \log(100) + \log(10) = \log(100 \whakareatia ki te 10) = \log(1000) \]
E mōhio ana tātou ka taea te tuhi i te 1000 hei \( 10^3 \), nō reira:
\[ \log(1000) = \log(10^3) \]
Mā te whakamahi i ngā ture o ngā logarithm:
\[ \log(10^3) = 3 \]
Nō reira, ko te uara o \( \log(100) + \log(10) \) he 3.
Te Huinga o ngā Taupū me ngā Tauira
I ētahi wā, me whakamahi ngātahi ngā taupū me ngā taupū whakarōpū hei whakaoti rapanga pāngarau.
Ngā Tauira Pātai Whakakotahi
Pātai 7:
Mena ko \( 2^x = 8 \), whakatauhia te uara o x.
Kōrero:
Hei whakatau i te uara o x, ka taea e tātou te tuhi i te 8 ki te āhua taupū me te pūtake 2.
\[ 8 = 2^3 \]
Nō reira ka pēnei te whārite:
\[ 2^x = 2^3 \]
Nā te mea he ōrite ngā turanga, me ōrite anō hoki ngā taupū.
\[ x = 3 \]
Nō reira, ko te uara o x ko te 3.
Pātai 8:
Whakatauhia te uara o \( \log_5 25 \).
Kōrero:
Hei whakatau i te uara o \( \log_5 25 \), me kimi e tātou te uara o te taupū e whakaputa ana i te 25 ina ko te pūtake he 5.
\[ 5^2 = 25 \]
Te tikanga,
\[ \log_5 25 = 2 \]
Nō reira, ko te uara o \( \log_5 25 \) he 2.
Pātai 9:
Mena ko te uara o te x ko te \( \log_2 ( x^2 ) = 6 \), whakatauhia te uara o te x.
Kōrero:
Hei whakatau i te uara o te x, ka taea e tātou te whakarerekē i te whārite taupūngao ki te āhua taupūpū.
\[ \log_2 ( x^2 ) = 6 \]
te tikanga,
\[ x^2 = 2^6 \]
\[ x^2 = 64 \]
Nō reira, me kimi e tātou te uara o x e tutuki ai te \( x^2 = 64 \).
\[ x = \sqrt{64} \]
\[ x = 8 \]
ranei
\[ x = -8 \]
Nō reira, ko te uara o x he 8, he -8 rānei.
Whakamutunga
He mea nui ngā taupū me ngā taurakitā i roto i te pāngarau. Mā te mārama me te mahi tika, ka taea e tātou te whakaoti ngāwari i ngā raruraru maha e pā ana ki ngā taupū me ngā taurakitā. Ko te tumanako ka āwhina ngā tauira i runga ake nei i a tātou ki te mārama ki ngā ariā taketake o ngā taupū me ngā taurakitā me pēhea te whakamahi i aua mea ki te whakaoti rapanga. Mā te mahi auau, ka mōhio ake tātou, ka matatau ake hoki ki te whakaoti rapanga pāngarau e pā ana ki ngā taupū me ngā taurakitā.
