Mocniny a odmocninysú jedným z najdôležitejších nástrojov v matematike,... Show more
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Updated Mar 18, 2026
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Základy Mocnín a Odmocnín
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Základné pojmy mocnín a odmocnín
Predstav si, že chceš vypočítať plochu štvorca so stranou 5 cm. Namiesto písania 5 × 5 použiješ jednoducho 5². Mocnina je skrátený zápis opakovaného násobenia rovnakého čísla.
Pri mocnine a^n je a základ mocniny (číslo, ktoré násobíme) a n je exponent (hovorí nám, koľkokrát násobíme). Napríklad 3⁴ = 3 × 3 × 3 × 3 = 81.
Odmocniny sú opak mocnín. Druhá odmocnina √a hľadá číslo, ktoré umocnené na druhú dá pôvodné číslo. Tretia odmocnina ∛a hľadá číslo pre tretiu mocninu.
Tip: Zapamätaj si, že zátvorky sú kľúčové! (-3)² = 9, ale -3² = -9.
Mocniny s rôznymi exponentmi
Kladný exponent znamená klasické násobenie - 3⁴ = 3 × 3 × 3 × 3 = 81. Pozor na záporné základy: ak je exponent párny, výsledok je kladný, ak nepárny, je záporný.
Nulový exponent má jednoduché pravidlo: akékoľvek číslo (okrem nuly) na nulu je 1. Takže 5⁰ = 1, (-100)⁰ = 1.
Záporný exponent znamená, že číslo "preklopíš" a exponent zmenis na kladný. 2⁻³ = 1/2³ = 1/8. Pri zlomkoch sa čitateľ a menovateľ vymenia: (2/3)⁻² = (3/2)² = 9/4.
Pozor: Záporný exponent neznamená záporný výsledok, ale prevrátené číslo!
Pravidlá pre počítanie s mocninami
Tieto vzorce ti ušetria množstvo času pri počítaní. Pri násobení mocnín s rovnakým základom sa exponenty sčítavajú: a^m · a^n = a^. Napríklad 2³ · 2⁵ = 2⁸.
Pri delení sa exponenty odčítavajú: a^m ÷ a^n = a^. Mocnina mocniny funguje tak, že exponenty násobíš: ^n = a^(m·n).
Mocnina súčinu a podielu sa rozkladá na jednotlivé mocniny: (a·b)^n = a^n · b^n a ^n = a^n/b^n. Toto ti pomôže pri zjednodušovaní zložitejších výrazov.
Skúška: Skús si 10³ · 10² = 10⁵ = 100 000. Funguje to!
Druhá odmocnina
Druhá odmocnina √a hľadá nezáporné číslo, ktoré umocnené na druhú dá a. Kľúčová podmienka: odmocnenec musí byť nezáporný (a ≥ 0) a výsledok je vždy nezáporný.
Napríklad √25 = 5, pretože 5² = 25. √0 = 0, ale √(-9) nemá riešenie v reálnych číslach.
Pravidlá pre odmocniny: √(a·b) = √a · √b a √ = √a/√b. Toto ti umožní zjednodušovať - √12 = √(4·3) = 2√3.
Častá chyba: √ ≠ √a + √b! Napríklad √(9+16) = √25 = 5, ale √9 + √16 = 3 + 4 = 7.
Tretia odmocnina a riešené príklady
Tretia odmocnina ∛a môže byť z akéhokoľvek čísla - kladného, záporného aj nuly. ∛8 = 2, ∛(-27) = -3, ∛0 = 0. Pravidlá sú podobné ako pri druhej odmocnine.
Pozrime si praktický príklad: ((2⁻¹)² · 4)⁰ + √144 - ∛(-64). Prvá časť je 1 (čokoľvek na nulu), √144 = 12, ∛(-64) = -4. Výsledok: 1 + 12 - (-4) = 17.
Pri zjednodušovaní ako √18 + √50 rozlož čísla: √18 = √(9·2) = 3√2, √50 = √(25·2) = 5√2. Potom sčítaj: 3√2 + 5√2 = 8√2.
Tip: Vždy sa snaž vytýkať úplné mocniny pre jednoduchší výsledok!
Zhrnutie a dôležité upozornenia
Kľúčové vzorce na zapamätanie: a^m · a^n = a^, a^m ÷ a^n = a^, ^n = a^(m·n), a⁰ = 1, a⁻n = 1/a^n. Pre odmocniny: √(a·b) = √a · √b.
Dôležité rozdiely: Druhá odmocnina len z nezáporných čísel s nezáporným výsledkom. Tretia odmocnina z akýchkoľvek čísel. Pozor na znamienka pri záporných základoch!
Častý omyl: Záporný exponent neznamená záporný výsledok, ale prevrátenie čísla. Tiež si dávaj pozor na poradie operácií - najprv zátvorky, potom mocniny a odmocniny.
Pre skúšku: Precvič si hlavne pravidlá pre mocniny a zjednodušovanie odmocnín!
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Is Knowunity really free of charge?
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Základy Mocnín a Odmocnín
Mocniny a odmocniny sú jedným z najdôležitejších nástrojov v matematike, ktoré ti pomôžu pri riešení úloh z geometrie, fyziky a ďalších predmetov. Namiesto písania dlhých násobení dokážeš používať skrátené zápisy a efektívne počítať.
Základné pojmy mocnín a odmocnín
Predstav si, že chceš vypočítať plochu štvorca so stranou 5 cm. Namiesto písania 5 × 5 použiješ jednoducho 5². Mocnina je skrátený zápis opakovaného násobenia rovnakého čísla.
Pri mocnine a^n je a základ mocniny (číslo, ktoré násobíme) a n je exponent (hovorí nám, koľkokrát násobíme). Napríklad 3⁴ = 3 × 3 × 3 × 3 = 81.
Odmocniny sú opak mocnín. Druhá odmocnina √a hľadá číslo, ktoré umocnené na druhú dá pôvodné číslo. Tretia odmocnina ∛a hľadá číslo pre tretiu mocninu.
Tip: Zapamätaj si, že zátvorky sú kľúčové! (-3)² = 9, ale -3² = -9.
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Mocniny s rôznymi exponentmi
Kladný exponent znamená klasické násobenie - 3⁴ = 3 × 3 × 3 × 3 = 81. Pozor na záporné základy: ak je exponent párny, výsledok je kladný, ak nepárny, je záporný.
Nulový exponent má jednoduché pravidlo: akékoľvek číslo (okrem nuly) na nulu je 1. Takže 5⁰ = 1, (-100)⁰ = 1.
Záporný exponent znamená, že číslo "preklopíš" a exponent zmenis na kladný. 2⁻³ = 1/2³ = 1/8. Pri zlomkoch sa čitateľ a menovateľ vymenia: (2/3)⁻² = (3/2)² = 9/4.
Pozor: Záporný exponent neznamená záporný výsledok, ale prevrátené číslo!
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Pravidlá pre počítanie s mocninami
Tieto vzorce ti ušetria množstvo času pri počítaní. Pri násobení mocnín s rovnakým základom sa exponenty sčítavajú: a^m · a^n = a^. Napríklad 2³ · 2⁵ = 2⁸.
Pri delení sa exponenty odčítavajú: a^m ÷ a^n = a^. Mocnina mocniny funguje tak, že exponenty násobíš: ^n = a^(m·n).
Mocnina súčinu a podielu sa rozkladá na jednotlivé mocniny: (a·b)^n = a^n · b^n a ^n = a^n/b^n. Toto ti pomôže pri zjednodušovaní zložitejších výrazov.
Skúška: Skús si 10³ · 10² = 10⁵ = 100 000. Funguje to!
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Druhá odmocnina
Druhá odmocnina √a hľadá nezáporné číslo, ktoré umocnené na druhú dá a. Kľúčová podmienka: odmocnenec musí byť nezáporný (a ≥ 0) a výsledok je vždy nezáporný.
Napríklad √25 = 5, pretože 5² = 25. √0 = 0, ale √(-9) nemá riešenie v reálnych číslach.
Pravidlá pre odmocniny: √(a·b) = √a · √b a √ = √a/√b. Toto ti umožní zjednodušovať - √12 = √(4·3) = 2√3.
Častá chyba: √ ≠ √a + √b! Napríklad √(9+16) = √25 = 5, ale √9 + √16 = 3 + 4 = 7.
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Tretia odmocnina a riešené príklady
Tretia odmocnina ∛a môže byť z akéhokoľvek čísla - kladného, záporného aj nuly. ∛8 = 2, ∛(-27) = -3, ∛0 = 0. Pravidlá sú podobné ako pri druhej odmocnine.
Pozrime si praktický príklad: ((2⁻¹)² · 4)⁰ + √144 - ∛(-64). Prvá časť je 1 (čokoľvek na nulu), √144 = 12, ∛(-64) = -4. Výsledok: 1 + 12 - (-4) = 17.
Pri zjednodušovaní ako √18 + √50 rozlož čísla: √18 = √(9·2) = 3√2, √50 = √(25·2) = 5√2. Potom sčítaj: 3√2 + 5√2 = 8√2.
Tip: Vždy sa snaž vytýkať úplné mocniny pre jednoduchší výsledok!
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Zhrnutie a dôležité upozornenia
Kľúčové vzorce na zapamätanie: a^m · a^n = a^, a^m ÷ a^n = a^, ^n = a^(m·n), a⁰ = 1, a⁻n = 1/a^n. Pre odmocniny: √(a·b) = √a · √b.
Dôležité rozdiely: Druhá odmocnina len z nezáporných čísel s nezáporným výsledkom. Tretia odmocnina z akýchkoľvek čísel. Pozor na znamienka pri záporných základoch!
Častý omyl: Záporný exponent neznamená záporný výsledok, ale prevrátenie čísla. Tiež si dávaj pozor na poradie operácií - najprv zátvorky, potom mocniny a odmocniny.
Pre skúšku: Precvič si hlavne pravidlá pre mocniny a zjednodušovanie odmocnín!
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We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user