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Nov 30, 2025
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Understanding Irrational Numbers - Lesson 8.10
mia!
@mdotti.ny
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Estimating Square Roots
Ever wondered how to estimate the value of a square root that isn't a perfect square? It's actually pretty simple!
To estimate a square root to the nearest integer, first identify the perfect squares that come before and after your number. For example, with √8, we know that 4 and 9 are perfect squares (2² and 3²), so 8 is between them. This means √8 is between 2 and 3.
To make the best estimate, see which perfect square is closer to your number. Since 8 is closer to 9 than to 4, we estimate √8 ≈ 3. For √83, we'd compare it to √81 (which equals 9) and √100 (which equals 10). Since 83 is closer to 81, our best estimate is 9.
Quick Tip: Draw a number line with the perfect squares marked if you're having trouble visualizing where your number falls. This makes it easier to see which whole number estimate makes the most sense!
Estimating Cube Roots
Cube roots work just like square roots, but we compare to perfect cubes instead of perfect squares.
To estimate a cube root like ∛320, first find the perfect cubes that sandwich your number. Since 6³ = 216 and 7³ = 343, we know 216 < 320 < 343. This means ∛320 is between 6 and 7. Since 320 is closer to 343, our best estimate is 7.
You can apply this same technique to any square root or cube root. For √35, we determine it's between √25 (5) and √36 (6). Since 35 is closer to 36, our best estimate is 6. For √170, it's between √169 (13) and √196 (14), with 13 being the better estimate since 170 is closer to 169.
Remember: When estimating, you're finding which whole number is closest to the actual value. You don't need to calculate the exact value!
Comparing Roots and Decimals
Sometimes you need to compare roots with decimal numbers to put them in order.
For cube roots like ∛62, we determine it's between ∛27 (3) and ∛64 (4). Since 62 is closer to 64, we estimate ∛62 ≈ 4. Similarly, for ∛25, we find it's between ∛8 (2) and ∛27 (3), with 3 being the better estimate.
When ordering a mix of roots and decimals, first convert the roots to decimal approximations. For example, √5 ≈ 2.236 and √6 ≈ 2.449. Now you can easily place them in order with other decimals like 2.3 and 2.5.
Pro Tip: Drawing a number line with all values marked can make ordering mixed numbers much easier - you can literally see which comes first!
Ordering Mixed Number Sets
Comparing sets with both square roots and decimals requires converting everything to a common format.
For a set like {√12, √15, 3.9, 3.5}, first approximate the square roots: √12 ≈ 3.464 and √15 ≈ 3.873. Now you can easily arrange all four numbers from least to greatest: √12, 3.5, √15, 3.9 (or approximately 3.464, 3.5, 3.873, 3.9).
This technique works for any set of mixed numbers. For {4.8, √26, 4.8, √22}, we find √22 ≈ 4.69 and √26 ≈ 5.10. Putting them in order: √22, 4.8, 4.8, √26 (or approximately 4.69, 4.8, 4.8, 5.10).
Did you know? Many real-world measurements result in square roots or cube roots. Engineers and scientists estimate these values all the time to make quick calculations!
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 THE SCHOOLGPT. 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 THE SCHOOLGPT. 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 Irrational Numbers - Lesson 8.10
mia!
@mdotti.ny
Estimating square roots and cube roots doesn't have to be complicated! In this lesson, you'll learn how to find approximate values for roots that aren't perfect squares or perfect cubes by comparing them to known values and making smart estimates.
Estimating Square Roots
Ever wondered how to estimate the value of a square root that isn't a perfect square? It's actually pretty simple!
To estimate a square root to the nearest integer, first identify the perfect squares that come before and after your number. For example, with √8, we know that 4 and 9 are perfect squares (2² and 3²), so 8 is between them. This means √8 is between 2 and 3.
To make the best estimate, see which perfect square is closer to your number. Since 8 is closer to 9 than to 4, we estimate √8 ≈ 3. For √83, we'd compare it to √81 (which equals 9) and √100 (which equals 10). Since 83 is closer to 81, our best estimate is 9.
Quick Tip: Draw a number line with the perfect squares marked if you're having trouble visualizing where your number falls. This makes it easier to see which whole number estimate makes the most sense!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Estimating Cube Roots
Cube roots work just like square roots, but we compare to perfect cubes instead of perfect squares.
To estimate a cube root like ∛320, first find the perfect cubes that sandwich your number. Since 6³ = 216 and 7³ = 343, we know 216 < 320 < 343. This means ∛320 is between 6 and 7. Since 320 is closer to 343, our best estimate is 7.
You can apply this same technique to any square root or cube root. For √35, we determine it's between √25 (5) and √36 (6). Since 35 is closer to 36, our best estimate is 6. For √170, it's between √169 (13) and √196 (14), with 13 being the better estimate since 170 is closer to 169.
Remember: When estimating, you're finding which whole number is closest to the actual value. You don't need to calculate the exact value!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Comparing Roots and Decimals
Sometimes you need to compare roots with decimal numbers to put them in order.
For cube roots like ∛62, we determine it's between ∛27 (3) and ∛64 (4). Since 62 is closer to 64, we estimate ∛62 ≈ 4. Similarly, for ∛25, we find it's between ∛8 (2) and ∛27 (3), with 3 being the better estimate.
When ordering a mix of roots and decimals, first convert the roots to decimal approximations. For example, √5 ≈ 2.236 and √6 ≈ 2.449. Now you can easily place them in order with other decimals like 2.3 and 2.5.
Pro Tip: Drawing a number line with all values marked can make ordering mixed numbers much easier - you can literally see which comes first!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Ordering Mixed Number Sets
Comparing sets with both square roots and decimals requires converting everything to a common format.
For a set like {√12, √15, 3.9, 3.5}, first approximate the square roots: √12 ≈ 3.464 and √15 ≈ 3.873. Now you can easily arrange all four numbers from least to greatest: √12, 3.5, √15, 3.9 (or approximately 3.464, 3.5, 3.873, 3.9).
This technique works for any set of mixed numbers. For {4.8, √26, 4.8, √22}, we find √22 ≈ 4.69 and √26 ≈ 5.10. Putting them in order: √22, 4.8, 4.8, √26 (or approximately 4.69, 4.8, 4.8, 5.10).
Did you know? Many real-world measurements result in square roots or cube roots. Engineers and scientists estimate these values all the time to make quick calculations!
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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4.9/5
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4.8/5
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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 THE SCHOOLGPT. 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 THE SCHOOLGPT. 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