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119
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11 Dec 2025
•
Julia Morgan
@juliamorgan_qhvr
Ever wondered how to expand expressions like (1+x)⁵ without multiplying... Show more
Think of Pascal's triangle as your expansion cheat sheet - each row gives you the coefficients you need! The triangle starts with 1 at the top, and every number below is the sum of the two numbers above it.
When you expand ⁿ, the coefficients come straight from Pascal's triangle. For example, ³ uses row 3: the coefficients 1, 3, 3, 1 give you 1 + 3x + 3x² + x³.
The binomial theorem formula is: ⁿ = aⁿ + (ⁿ₁)aⁿ⁻¹b + (ⁿ₂)aⁿ⁻²b² + ... + bⁿ. Don't let the notation scare you - those brackets are just Pascal's triangle numbers in disguise!
Quick Tip: The powers of the first term decrease whilst the powers of the second term increase, but they always add up to n.
Now let's tackle trickier expressions like ⁴ or ⁵. The process stays the same, but you need to be careful with your arithmetic and signs.
For ⁴, use Pascal's triangle coefficients 1, 4, 6, 4, 1, but remember that 2a gets raised to different powers. So (2a)² becomes 4a², (2a)³ becomes 8a³, and so on.
With expressions involving negative terms like ⁵, watch your signs carefully! The pattern alternates: positive, negative, positive, negative. Each odd power of the negative term flips the sign.
The key is staying organised - write out each term step by step, calculate the coefficient from Pascal's triangle, then work out the powers of each part separately.
Watch Out: Negative signs can be tricky - double-check that your alternating pattern is correct, especially with higher powers.
You're getting the hang of this! Let's expand expressions like ³ or ⁵. The trick is treating each part of the binomial carefully.
For ³, your Pascal's triangle gives 1, 3, 3, 1. But when you calculate (3b)², you get 9b², and (3b)³ gives 27b³. Don't forget to multiply the coefficient by the number raised to that power!
With ⁵, you're working with coefficients 1, 5, 10, 10, 5, 1 from Pascal's triangle. Then (2s)² becomes 4s², (2s)³ becomes 8s³, and so on.
The pattern stays consistent - use Pascal's triangle for the main coefficients, then carefully calculate what happens when you raise each term to its respective power.
Pro Tip: Always double-check your arithmetic when dealing with coefficients - it's easy to make calculation errors that throw off your entire answer.
Ready for the challenge? Let's tackle expressions with fractions and negative terms like ⁶ or ³.
For ⁶, you'll use Pascal's triangle row 6: 1, 6, 15, 20, 15, 6, 1. Remember that ² gives +4c² (positive because you're squaring a negative), but ³ gives -8c³ (negative because it's an odd power).
Fractional expressions like ³ follow the same rules. Your terms become 1 + 3 + 3² + ³, which simplifies to 1 + 3/x + 3/x² + 1/x³.
The key is patience and systematic working. Write each term out fully before simplifying, and always check your signs and fractions twice.
Challenge Accepted: These harder expansions are exactly what examiners love to test - master them and you'll stand out from the crowd!
Here's where the maths gets really clever! Factorials (written as n!) count the number of ways to arrange n objects. So 4! = 4 × 3 × 2 × 1 = 24.
The combination notation ⁿCᵣ or (n choose r) gives you those Pascal's triangle coefficients directly. The formula is ⁿCᵣ = n!/!r!. This tells you how many ways you can choose r items from n items.
For example, ⁴C₂ = 4!/(2!×2!) = 24/(2×2) = 6. This matches the Pascal's triangle coefficient in row 4, position 2!
Understanding this connection makes the binomial theorem much more powerful - you can find any coefficient without drawing out Pascal's triangle.
Mind Blown: Those Pascal's triangle numbers are actually counting combinations - maths is beautifully connected!
Now you can calculate combination coefficients for any binomial expansion, even with massive numbers! Use the formula ⁿCᵣ = n!/!r! systematically.
For ¹⁰C₄, you get 10!/(6!×4!) = 210. Notice that ⁿCᵣ = ⁿCₙ₋ᵣ - this symmetry can save you calculation time. So ¹⁰C₄ = ¹⁰C₆.
Some useful patterns to remember: ⁿC₀ = 1, ⁿC₁ = n, and ⁿCₙ₋₁ = n. These pop up regularly in exam questions.
For expressions like ⁿC₃, you can write this as n/6 without calculating full factorials. This makes algebraic problems much more manageable.
Exam Success: Learning these patterns and shortcuts will save you precious time in exams and help you tackle the trickiest questions with confidence.
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
App Store
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Julia Morgan
@juliamorgan_qhvr
Ever wondered how to expand expressions like (1+x)⁵ without multiplying everything out the long way? The Binomial Theorem is your mathematical shortcut that uses Pascal's triangle and factorials to expand any binomial expression quickly and accurately.
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Think of Pascal's triangle as your expansion cheat sheet - each row gives you the coefficients you need! The triangle starts with 1 at the top, and every number below is the sum of the two numbers above it.
When you expand ⁿ, the coefficients come straight from Pascal's triangle. For example, ³ uses row 3: the coefficients 1, 3, 3, 1 give you 1 + 3x + 3x² + x³.
The binomial theorem formula is: ⁿ = aⁿ + (ⁿ₁)aⁿ⁻¹b + (ⁿ₂)aⁿ⁻²b² + ... + bⁿ. Don't let the notation scare you - those brackets are just Pascal's triangle numbers in disguise!
Quick Tip: The powers of the first term decrease whilst the powers of the second term increase, but they always add up to n.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Now let's tackle trickier expressions like ⁴ or ⁵. The process stays the same, but you need to be careful with your arithmetic and signs.
For ⁴, use Pascal's triangle coefficients 1, 4, 6, 4, 1, but remember that 2a gets raised to different powers. So (2a)² becomes 4a², (2a)³ becomes 8a³, and so on.
With expressions involving negative terms like ⁵, watch your signs carefully! The pattern alternates: positive, negative, positive, negative. Each odd power of the negative term flips the sign.
The key is staying organised - write out each term step by step, calculate the coefficient from Pascal's triangle, then work out the powers of each part separately.
Watch Out: Negative signs can be tricky - double-check that your alternating pattern is correct, especially with higher powers.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
You're getting the hang of this! Let's expand expressions like ³ or ⁵. The trick is treating each part of the binomial carefully.
For ³, your Pascal's triangle gives 1, 3, 3, 1. But when you calculate (3b)², you get 9b², and (3b)³ gives 27b³. Don't forget to multiply the coefficient by the number raised to that power!
With ⁵, you're working with coefficients 1, 5, 10, 10, 5, 1 from Pascal's triangle. Then (2s)² becomes 4s², (2s)³ becomes 8s³, and so on.
The pattern stays consistent - use Pascal's triangle for the main coefficients, then carefully calculate what happens when you raise each term to its respective power.
Pro Tip: Always double-check your arithmetic when dealing with coefficients - it's easy to make calculation errors that throw off your entire answer.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Ready for the challenge? Let's tackle expressions with fractions and negative terms like ⁶ or ³.
For ⁶, you'll use Pascal's triangle row 6: 1, 6, 15, 20, 15, 6, 1. Remember that ² gives +4c² (positive because you're squaring a negative), but ³ gives -8c³ (negative because it's an odd power).
Fractional expressions like ³ follow the same rules. Your terms become 1 + 3 + 3² + ³, which simplifies to 1 + 3/x + 3/x² + 1/x³.
The key is patience and systematic working. Write each term out fully before simplifying, and always check your signs and fractions twice.
Challenge Accepted: These harder expansions are exactly what examiners love to test - master them and you'll stand out from the crowd!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Here's where the maths gets really clever! Factorials (written as n!) count the number of ways to arrange n objects. So 4! = 4 × 3 × 2 × 1 = 24.
The combination notation ⁿCᵣ or (n choose r) gives you those Pascal's triangle coefficients directly. The formula is ⁿCᵣ = n!/!r!. This tells you how many ways you can choose r items from n items.
For example, ⁴C₂ = 4!/(2!×2!) = 24/(2×2) = 6. This matches the Pascal's triangle coefficient in row 4, position 2!
Understanding this connection makes the binomial theorem much more powerful - you can find any coefficient without drawing out Pascal's triangle.
Mind Blown: Those Pascal's triangle numbers are actually counting combinations - maths is beautifully connected!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Now you can calculate combination coefficients for any binomial expansion, even with massive numbers! Use the formula ⁿCᵣ = n!/!r! systematically.
For ¹⁰C₄, you get 10!/(6!×4!) = 210. Notice that ⁿCᵣ = ⁿCₙ₋ᵣ - this symmetry can save you calculation time. So ¹⁰C₄ = ¹⁰C₆.
Some useful patterns to remember: ⁿC₀ = 1, ⁿC₁ = n, and ⁿCₙ₋₁ = n. These pop up regularly in exam questions.
For expressions like ⁿC₃, you can write this as n/6 without calculating full factorials. This makes algebraic problems much more manageable.
Exam Success: Learning these patterns and shortcuts will save you precious time in exams and help you tackle the trickiest questions with confidence.
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
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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MathsExplore essential concepts of indices, including simple index rules, negative and fractional powers, and techniques for evaluating exponents. This summary provides clear examples and step-by-step methods to enhance your understanding of powers and roots in mathematics.
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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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Sociology