Ke Hoʻākāka nei i nā ʻAno Kumu: Kūkākūkā ʻana i nā Pilikia Hoʻohālike
ʻO ka hoʻākāka ʻana i nā radicals he mākaukau kumu ia i ka algebra he mea nui e aʻo ai. Hoʻololi kēia kaʻina hana i nā hakina me nā radicals i loko o ka denominator i kahi ʻano kūpono. Ma kēia ʻatikala, e uhi mākou i nā manaʻo kumu, nā pono, a hāʻawi i kekahi mau pilikia hoʻohālike a me nā hoʻonā e pili ana i ka hoʻākāka ʻana i nā radicals.
Nā Manaʻo Kumu o ka Hoʻākāka ʻana i nā ʻAno Aʻa
ʻO ka hoʻākāka ʻana i kahi radical ke ʻano o ka hoʻololi ʻana i ka denominator i kahi hakina me kahi radical i ʻole he radical i loko o ka denominator. ʻO ke kumu nui a mākou e hana ai i kēia, ʻo ia ka hoʻomaʻalahi ʻana i nā helu a me ka maʻalahi o ka heluhelu ʻana a me ka hoʻohālikelike ʻana i nā waiwai o nā hōʻike.
Nā Pōmaikaʻi o ka Hoʻomaopopo ʻana i ke ʻAno o ke Aʻa
1. Hoʻomaʻalahi i ka helu ʻana: ʻOi aku ka maʻalahi o ka loiloi ʻana i nā hakina me nā denominators me ke aʻa ʻole ma ka lima a me ka hoʻohana ʻana i ka mīkini helu.
2. Ke hōʻoia ʻana i ke kūlike: Pono nā puke aʻo a me nā kūlana hoʻokolohua he nui e hōʻike ʻia nā hapa ma nā ʻano maʻalahi a kūpono hoʻi.
3. Ke Hoʻohālikelike ʻana i nā Waiwai: ʻOi aku ka maʻalahi o ka hoʻohālikelike ʻana i nā ʻano kūpono kekahi i kekahi no ka mea ʻoi aku ka maopopo o kā lākou mau waiwai.
Nā Hana e Hoʻomaopopo ai i ke ʻAno o ke Aʻa
No ka hoʻākāka ʻana i ke ʻano kumu i loko o ka denominator, pono mākou e hoʻonui i ka numerator a me ka denominator me ke ʻano kūpono i lilo ai ka denominator i helu rational. Eia nā ʻanuʻu:
1. E ʻike i nā aʻa i loko o ka Denominator: E hōʻoia i ke ʻano o nā aʻa i kahi hakina e pono ai ke wehewehe ʻana.
2. Hoʻonui me ke ʻano kūpono: ʻO ke ʻano a mākou e hoʻohana ai e pili ana i ke ʻano o ka radical i loko o ka denominator. ʻEkolu mau ʻano maʻamau e pono e hoʻomaopopo ʻia:
– Nā ʻano maʻalahi e like me \(\sqrt{a}\).
– Nā ʻano binomial e like me \(\sqrt{a} + b\) a i ʻole \(\sqrt{a} – b\).
– Nā aʻa mana kiʻekiʻe e like me \(\sqrt[3]{a}\).
Nā Nīnau Laʻana a me ke Kūkākūkā
Laʻana 1: Ke Hoʻākāka nei i ka Denominator me nā Aʻa Maʻalahi
Nīnau:
\[ \frac{5}{\sqrt{3}} \]
Kūkākūkā:
1. E ʻike i nā aʻa i loko o ka Denominator: ʻO ka denominator ʻo \(\sqrt{3}\).
2. Hoʻonui me ke ʻAno Kūpono: Makemake mākou e wehe i ka radical mai ka denominator ma ka hoʻonui ʻana i ka numerator a me ka denominator me \(\sqrt{3}\).
\[
\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{5\sqrt{3}}{3}
\]
No laila, \(\frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}\).
Laʻana 2: Ke Hoʻākāka nei i ka Denominator me nā Aʻa Binomial
Nīnau:
\[ \frac{4}{\sqrt{2} + 1} \]
Kūkākūkā:
1. E ʻike i nā aʻa i loko o ka Denominator: Aia ka denominator ma ke ʻano binomial, ʻo ia hoʻi \(\sqrt{2} + 1\).
2. Hoʻonui me ke ʻano kūpono: Hoʻohana mākou i ka hui hui o \(\sqrt{2} + 1\), ʻo ia hoʻi ʻo \(\sqrt{2} – 1\).
\[
\frac{4}{\sqrt{2} + 1} \times \frac{\sqrt{2} – 1}{\sqrt{2} – 1} = \frac{4(\sqrt{2} – 1)}{(\sqrt{2} + 1)(\sqrt{2} – 1)}
\]
3. Hoʻomaʻalahi i ka Denominator: E hoʻohana i nā ʻike algebraic e hoʻomaʻalahi i ka denominator:
\[
(\sqrt{2} + 1)(\sqrt{2} – 1) = (\sqrt{2})^2 – (1)^2 = 2 – 1 = 1
\]
No laila, lilo ka hapa i:
\[
\frac{4(\sqrt{2} – 1)}{1} = 4\sqrt{2} – 4
\]
No laila, \(\frac{4}{\sqrt{2} + 1} = 4\sqrt{2} – 4\).
Laʻana 3: Ke Hoʻākāka nei i ka Denominator me nā Aʻa Cube
Nīnau:
\[ \frac{7}{\sqrt[3]{4}} \]
Kūkākūkā:
1. E ʻike i nā aʻa i loko o ka Denominator: ʻO ka denominator ʻo \(\sqrt[3]{4}\).
2. Hoʻonui me ke ʻano kūpono: E hoʻohana i ka \((\sqrt[3]{4})^2\) no ka mea \(\sqrt[3]{4} \times (\sqrt[3]{4})^2 = 4\).
\[
\frac{7}{\sqrt[3]{4}} \times \frac{(\sqrt[3]{4})^2}{(\sqrt[3]{4})^2} = \frac{7(\sqrt[3]{4})^2}{4}
\]
Waiho mākou iā \((\sqrt[3]{4})^2\) ma ke ʻano kumu cube no ka mea ʻo kēia ke ʻano i ʻae pinepine ʻia:
\[
\frac{7 \cdot \sqrt[3]{16}}{4}
\]
No laila, \(\frac{7}{\sqrt[3]{4}} = \frac{7 \sqrt[3]{16}}{4}\).
Laʻana 4: Ke Hoʻākāka nei i ke ʻAno Kumu me nā Hoʻomaʻalahi Hou Aʻe
Nīnau:
\[ \frac{2\sqrt{5}}{\sqrt{3} + \sqrt{2}} \]
Kūkākūkā:
1. E ʻike i nā aʻa i loko o ka Denominator: ʻO ka denominator ʻo \(\sqrt{3} + \sqrt{2}\).
2. Hoʻonui me ke ʻano kūpono: E hoʻohana i ka hui pū ʻana o \(\sqrt{3} + \sqrt{2}\), ʻo ia hoʻi \(\sqrt{3} – \sqrt{2}\).
\[
\frac{2\sqrt{5}}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} – \sqrt{2}}{\sqrt{3} – \sqrt{2}} = \frac{2\sqrt{5}(\sqrt{3} – \sqrt{2})}{(\sqrt{3})^2 – (\sqrt{2})^2}
\]
3. E hoʻomaʻalahi i ka Denominator:
\[
(\sqrt{3})^2 – (\sqrt{2})^2 = 3 – 2 = 1
\]
No laila, lilo ka hapa i:
\[
2\sqrt{5}(\sqrt{3} – \sqrt{2}) = 2\sqrt{15} – 2\sqrt{10}
\]
No laila, \(\frac{2\sqrt{5}}{\sqrt{3} + \sqrt{2}} = 2\sqrt{15} – 2\sqrt{10}\).
Ka hopena
He mākaukau makemakika koʻikoʻi ka hoʻomaopopo ʻana i nā ʻano aʻa e aʻo ai. ʻAʻole wale kēia e kōkua i ka hoʻomaʻalahi ʻana i nā helu akā e maʻalahi hoʻi ka loiloi a me ka hoʻohālikelike ʻana i nā waiwai. Ma o nā nīnau hoʻohālike a me ke kūkākūkā ʻana ma luna, hiki iā mākou ke hoʻomaopopo i nā ʻano hana like ʻole i hoʻohana ʻia e hoʻomaopopo i ke ʻano aʻa i loko o ka denominator, inā he ʻano maʻalahi, binomial, a i ʻole nā aʻa mana kiʻekiʻe. Me ka hoʻomaʻamaʻa hou aku, e ʻoi aku ka akamai a me ka wikiwiki o mākou i ka hoʻomaopopo ʻana i nā ʻano aʻa.
