Powers and roots are mathematical shortcuts that make calculations much... Show more
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Updated Mar 17, 2026
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Understanding Powers, Squares, Cubes, and Roots
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Getting Started with Powers and Roots
Ever wondered why we write 4³ instead of 4 × 4 × 4? Powers are basically a clever shortcut for repeated multiplication, making your maths look much tidier. The base is the number being multiplied (like the 4), and the index (also called power or exponent) is that small number up top telling you how many times to multiply.
When a number has a power of 2, we call it squared - like 3² is "3 squared". This name comes from finding the area of a square! Similarly, a power of 3 is called cubed because it's how you calculate a cube's volume.
Square numbers are what you get when you multiply any whole number by itself. For example, 9 is a square number because 3 × 3 = 9. These will pop up everywhere in your exams, so they're worth remembering!
Quick Tip: Memorising the first 12 square numbers will make your exam much faster and easier.
Working with Square Numbers and Cubes
You'll definitely want to memorise these square numbers for tests: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, and 12² = 144. Trust me, knowing these off by heart will save you loads of time.
Cube numbers work similarly but with three multiplications instead of two. The first few are: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, and 5³ = 125.
Here's something interesting - 64 is both a square number (8²) and a cube number (4³). That's definitely worth remembering for trick questions!
Remember: Don't confuse 6² with 6 × 2! The first equals 36, the second equals 12 - completely different answers.
Understanding Square Roots
Square roots are the complete opposite of squaring - they're like mathematical detective work. When you see √25, it's asking "what number multiplied by itself gives 25?" The answer is 5, because 5 × 5 = 25.
Think of it like this: if you know a square has an area of 25 cm², the square root helps you find that each side is 5 cm long. That's why we use the square root symbol √.
Perfect squares are numbers whose square roots are whole numbers, like 1, 4, 9, 16, and 25. These are the easiest ones to work with because there's no messy decimals involved.
Visual Tip: Imagine a square with area 25 cm² - the square root finds the length of each side (5 cm).
Worked Examples You Can Master
Let's tackle 9²: The base is 9, the index is 2, so we multiply 9 by itself once. That's 9 × 9 = 81. See how straightforward that is?
For 4³, we've got base 4 and index 3, meaning three 4s multiplied together: 4 × 4 × 4. First do 4 × 4 = 16, then 16 × 4 = 64. Breaking it into steps makes it much easier.
Finding √64 means asking "what number times itself equals 64?" Work through your square numbers: 6² = 36 (too small), 7² = 49 (getting closer), 8² = 64 (perfect!). So √64 = 8.
Exam Strategy: Show your working step by step - even if you use a calculator to check, you need to demonstrate your method.
Essential Tips for Test Success
Any number to the power of 1 is just itself - so 8¹ = 8. This might seem obvious, but it catches people out in exams when they overthink it.
The calculator's √ button is handy for checking answers, but always show your working. Examiners want to see that you understand the process, not just that you can press buttons.
Your key takeaways: powers use a base and index (like 5²), squared means power of 2, cubed means power of 3, and square roots reverse the squaring process. Master these basics and you'll smash any powers and roots question.
Final Reminder: Perfect squares (1, 4, 9, 16, 25...) are your best friends - they have nice, neat whole number square roots.
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.
Most popular content in Mathematics
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5
App Store
4.7/5
Google Play
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
Understanding Powers, Squares, Cubes, and Roots
Powers and roots are mathematical shortcuts that make calculations much easier and neater. Powers let you write repeated multiplication in a compact way, whilst roots help you work backwards to find the original number that was multiplied.
Getting Started with Powers and Roots
Ever wondered why we write 4³ instead of 4 × 4 × 4? Powers are basically a clever shortcut for repeated multiplication, making your maths look much tidier. The base is the number being multiplied (like the 4), and the index (also called power or exponent) is that small number up top telling you how many times to multiply.
When a number has a power of 2, we call it squared - like 3² is "3 squared". This name comes from finding the area of a square! Similarly, a power of 3 is called cubed because it's how you calculate a cube's volume.
Square numbers are what you get when you multiply any whole number by itself. For example, 9 is a square number because 3 × 3 = 9. These will pop up everywhere in your exams, so they're worth remembering!
Quick Tip: Memorising the first 12 square numbers will make your exam much faster and easier.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Working with Square Numbers and Cubes
You'll definitely want to memorise these square numbers for tests: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, and 12² = 144. Trust me, knowing these off by heart will save you loads of time.
Cube numbers work similarly but with three multiplications instead of two. The first few are: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, and 5³ = 125.
Here's something interesting - 64 is both a square number (8²) and a cube number (4³). That's definitely worth remembering for trick questions!
Remember: Don't confuse 6² with 6 × 2! The first equals 36, the second equals 12 - completely different answers.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Understanding Square Roots
Square roots are the complete opposite of squaring - they're like mathematical detective work. When you see √25, it's asking "what number multiplied by itself gives 25?" The answer is 5, because 5 × 5 = 25.
Think of it like this: if you know a square has an area of 25 cm², the square root helps you find that each side is 5 cm long. That's why we use the square root symbol √.
Perfect squares are numbers whose square roots are whole numbers, like 1, 4, 9, 16, and 25. These are the easiest ones to work with because there's no messy decimals involved.
Visual Tip: Imagine a square with area 25 cm² - the square root finds the length of each side (5 cm).
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Worked Examples You Can Master
Let's tackle 9²: The base is 9, the index is 2, so we multiply 9 by itself once. That's 9 × 9 = 81. See how straightforward that is?
For 4³, we've got base 4 and index 3, meaning three 4s multiplied together: 4 × 4 × 4. First do 4 × 4 = 16, then 16 × 4 = 64. Breaking it into steps makes it much easier.
Finding √64 means asking "what number times itself equals 64?" Work through your square numbers: 6² = 36 (too small), 7² = 49 (getting closer), 8² = 64 (perfect!). So √64 = 8.
Exam Strategy: Show your working step by step - even if you use a calculator to check, you need to demonstrate your method.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Essential Tips for Test Success
Any number to the power of 1 is just itself - so 8¹ = 8. This might seem obvious, but it catches people out in exams when they overthink it.
The calculator's √ button is handy for checking answers, but always show your working. Examiners want to see that you understand the process, not just that you can press buttons.
Your key takeaways: powers use a base and index (like 5²), squared means power of 2, cubed means power of 3, and square roots reverse the squaring process. Master these basics and you'll smash any powers and roots question.
Final Reminder: Perfect squares (1, 4, 9, 16, 25...) are your best friends - they have nice, neat whole number square roots.
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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Most popular content in Mathematics
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5
App Store
4.7/5
Google Play
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