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Algebra 1
Dec 2, 2025
53
8 pages
Neil Trivedi - MyEdSpace @neildoesmaths
Index laws are essential mathematical rules that help you work with powers and roots efficiently. These laws show... Show more
Ever wondered why mathematicians love shortcuts? Index laws are your secret weapon for working with powers quickly and accurately. When you see something like x³, the small number (called an index or power) tells you how many times to multiply x by itself.
There are nine fundamental laws that govern how indices behave. Once you master these rules, you'll find algebra becomes much more manageable. The plural of index is indices, so don't be confused when you see both terms used.
Quick Tip Think of index laws as mathematical shortcuts - they help you avoid writing out long multiplication chains every time you work with powers.
These laws work for any number or variable, making them incredibly useful for solving equations, simplifying expressions, and tackling real-world problems.
When you multiply powers with the same base, you simply add the indices together x^a × x^b = x^. This makes perfect sense when you think about what multiplication actually means.
For example, x³ × x² becomes x⁵ because you're multiplying (x × x × x) by (x × x), giving you five x's multiplied together. The key is that the base must be the same - you can't use this rule with different variables.
This law also works brilliantly with coefficients (numbers in front). When simplifying 3x⁴y × 2xy², you multiply the numbers (3 × 2 = 6) and add the indices for each variable separately, giving you 6x⁵y³.
Real-World Connection This law is perfect for finding areas of rectangles when the sides are given as algebraic expressions - just multiply length by width and add the indices.
Division with indices follows the opposite pattern to multiplication x^a ÷ x^b = x^. You subtract the bottom index from the top index, which makes sense when you think about cancelling out common factors.
Take x⁶ ÷ x² as an example. When written as a fraction, you can cancel out two x's from the top and bottom, leaving you with x⁴. This subtraction rule saves you from writing out all those x's every time.
The same principle applies to more complex expressions like 10x³y² ÷ 5xy. First, divide the numbers (10 ÷ 5 = 2), then subtract indices for each variable separately, giving you 2x²y.
Watch Out When the bottom index is larger than the top index, you'll end up with a negative power - but don't panic, there are laws to handle those too!
When you have a power raised to another power, like ^b, you multiply the indices together ^b = x^(ab). This rule helps you deal with brackets containing powers efficiently.
Think of (y⁴)⁵ as y⁴ multiplied by itself five times. Using Law 1, you'd add 4+4+4+4+4 = 20, giving you y²⁰. The shortcut is simply 4 × 5 = 20, so (y⁴)⁵ = y²⁰.
This law becomes particularly useful with algebraic expressions. For example, (x⁶)^ becomes x^ = x^. When there's a coefficient like (2x⁴)², remember to square both the number and the variable part separately.
Pro Tip Always multiply the indices when you see brackets with powers - this law will save you tons of time in algebra questions.
Here's where things get interesting! Any number to the power of zero equals 1 x⁰ = 1. This might seem strange, but it's a fundamental rule that keeps mathematics consistent. So 4x⁰ simply equals 4 × 1 = 4.
Negative powers represent reciprocals (flipped fractions). When you see x⁻¹, it equals 1/x. This is incredibly useful for simplifying division problems where the bottom index is larger than the top index.
For instance, 8x² ÷ 4x³ gives you 2x⁻¹, which you can rewrite as 2/x. The negative index tells you the variable has "moved" to the denominator of the fraction.
Memory Trick Think of negative indices as "upside-down" powers - they flip the number into the denominator of a fraction.
Law 6 extends the reciprocal idea x⁻ⁿ = 1/xⁿ. So 12x⁻³ becomes 12/x³. This law helps you convert between negative indices and fractions effortlessly.
Law 7 deals with fractions raised to powers ⁿ = xⁿ/yⁿ. You simply raise both the numerator and denominator to the same power. For example, (3/4)³ = 27/64.
When you combine negative powers with fractions, something magical happens. Take (7/5)⁻² the negative power flips the fraction first, giving you (5/7)², which then equals 25/49.
Key Insight Negative powers with fractions always flip the fraction first, then apply the positive power - this two-step process will never let you down.
Fractional indices represent roots x^ = ⁿ√x. This connection between powers and roots opens up a whole new world of mathematical shortcuts. The denominator of the fraction tells you which root to take.
For example, 64^(1/2) means the square root of 64, which equals 8. Similarly, 16^(1/4) means the fourth root of 16. Since 2 × 2 × 2 × 2 = 16, the fourth root is 2.
When you see just a root symbol without a number (like √), it's automatically a square root. This is similar to how we write x instead of x¹ - the "2" is understood to be there.
Calculator Tip Your calculator's power button can handle fractional indices, making it easy to check your root calculations.
The final law combines everything x^ = (ⁿ√x)ᵐ. You take the nth root first, then raise the result to the power m. This two-step process makes even complex-looking expressions manageable.
Let's tackle 36^(3/2) first find the square root of 36 (which is 6), then cube it to get 6³ = 216. The fraction 3/2 tells you everything - take the square root, then cube the result.
Negative fractional powers add an extra flip. For 9^(-5/2), you first find 1/, then work out (√9)⁵ = 3⁵ = 243, giving you 1/243 as your final answer.
Success Strategy Break fractional powers into two clear steps - find the root first (denominator), then apply the power (numerator). With negative powers, remember to flip at the end.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
1
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
PEMDAS (GEMDAS)
Algebra 1App 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
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
Neil Trivedi - MyEdSpace
@neildoesmaths
Index laws are essential mathematical rules that help you work with powers and roots efficiently. These laws show you how to multiply, divide, and manipulate expressions with indices, making complex calculations much simpler and helping you tackle GCSE maths problems... Show more
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Ever wondered why mathematicians love shortcuts? Index laws are your secret weapon for working with powers quickly and accurately. When you see something like x³, the small number (called an index or power) tells you how many times to multiply x by itself.
There are nine fundamental laws that govern how indices behave. Once you master these rules, you'll find algebra becomes much more manageable. The plural of index is indices, so don't be confused when you see both terms used.
Quick Tip: Think of index laws as mathematical shortcuts - they help you avoid writing out long multiplication chains every time you work with powers.
These laws work for any number or variable, making them incredibly useful for solving equations, simplifying expressions, and tackling real-world problems.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When you multiply powers with the same base, you simply add the indices together: x^a × x^b = x^. This makes perfect sense when you think about what multiplication actually means.
For example, x³ × x² becomes x⁵ because you're multiplying (x × x × x) by (x × x), giving you five x's multiplied together. The key is that the base must be the same - you can't use this rule with different variables.
This law also works brilliantly with coefficients (numbers in front). When simplifying 3x⁴y × 2xy², you multiply the numbers (3 × 2 = 6) and add the indices for each variable separately, giving you 6x⁵y³.
Real-World Connection: This law is perfect for finding areas of rectangles when the sides are given as algebraic expressions - just multiply length by width and add the indices.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Division with indices follows the opposite pattern to multiplication: x^a ÷ x^b = x^. You subtract the bottom index from the top index, which makes sense when you think about cancelling out common factors.
Take x⁶ ÷ x² as an example. When written as a fraction, you can cancel out two x's from the top and bottom, leaving you with x⁴. This subtraction rule saves you from writing out all those x's every time.
The same principle applies to more complex expressions like 10x³y² ÷ 5xy. First, divide the numbers (10 ÷ 5 = 2), then subtract indices for each variable separately, giving you 2x²y.
Watch Out: When the bottom index is larger than the top index, you'll end up with a negative power - but don't panic, there are laws to handle those too!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When you have a power raised to another power, like ^b, you multiply the indices together: ^b = x^(ab). This rule helps you deal with brackets containing powers efficiently.
Think of (y⁴)⁵ as y⁴ multiplied by itself five times. Using Law 1, you'd add 4+4+4+4+4 = 20, giving you y²⁰. The shortcut is simply 4 × 5 = 20, so (y⁴)⁵ = y²⁰.
This law becomes particularly useful with algebraic expressions. For example, (x⁶)^ becomes x^ = x^. When there's a coefficient like (2x⁴)², remember to square both the number and the variable part separately.
Pro Tip: Always multiply the indices when you see brackets with powers - this law will save you tons of time in algebra questions.
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 things get interesting! Any number to the power of zero equals 1: x⁰ = 1. This might seem strange, but it's a fundamental rule that keeps mathematics consistent. So 4x⁰ simply equals 4 × 1 = 4.
Negative powers represent reciprocals (flipped fractions). When you see x⁻¹, it equals 1/x. This is incredibly useful for simplifying division problems where the bottom index is larger than the top index.
For instance, 8x² ÷ 4x³ gives you 2x⁻¹, which you can rewrite as 2/x. The negative index tells you the variable has "moved" to the denominator of the fraction.
Memory Trick: Think of negative indices as "upside-down" powers - they flip the number into the denominator of a fraction.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Law 6 extends the reciprocal idea: x⁻ⁿ = 1/xⁿ. So 12x⁻³ becomes 12/x³. This law helps you convert between negative indices and fractions effortlessly.
Law 7 deals with fractions raised to powers: ⁿ = xⁿ/yⁿ. You simply raise both the numerator and denominator to the same power. For example, (3/4)³ = 27/64.
When you combine negative powers with fractions, something magical happens. Take (7/5)⁻²: the negative power flips the fraction first, giving you (5/7)², which then equals 25/49.
Key Insight: Negative powers with fractions always flip the fraction first, then apply the positive power - this two-step process will never let you down.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Fractional indices represent roots: x^ = ⁿ√x. This connection between powers and roots opens up a whole new world of mathematical shortcuts. The denominator of the fraction tells you which root to take.
For example, 64^(1/2) means the square root of 64, which equals 8. Similarly, 16^(1/4) means the fourth root of 16. Since 2 × 2 × 2 × 2 = 16, the fourth root is 2.
When you see just a root symbol without a number (like √), it's automatically a square root. This is similar to how we write x instead of x¹ - the "2" is understood to be there.
Calculator Tip: Your calculator's power button can handle fractional indices, making it easy to check your root calculations.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The final law combines everything: x^ = (ⁿ√x)ᵐ. You take the nth root first, then raise the result to the power m. This two-step process makes even complex-looking expressions manageable.
Let's tackle 36^(3/2): first find the square root of 36 (which is 6), then cube it to get 6³ = 216. The fraction 3/2 tells you everything - take the square root, then cube the result.
Negative fractional powers add an extra flip. For 9^(-5/2), you first find 1/, then work out (√9)⁵ = 3⁵ = 243, giving you 1/243 as your final answer.
Success Strategy: Break fractional powers into two clear steps - find the root first (denominator), then apply the power (numerator). With negative powers, remember to flip at the end.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
1
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
PEMDAS (GEMDAS)
Algebra 1App 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
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
