Laws of Integral Indices: Your Ultimate Study Guide!
Hello there! Welcome to the world of indices. It might sound a bit fancy, but it's really just a super-fast way of writing down repeated multiplication. Think of it as a maths shortcut! In these notes, we're going to unlock the secrets of indices (also called powers or exponents). You'll learn the rules that make them work, how to handle weird-looking things like negative and zero powers, and even see how they're used to write gigantic numbers in science.
Don't worry if this seems tricky at first, we'll break it all down step-by-step. Let's get started!
First Things First: What Are Indices?
An index tells you how many times to multiply a number by itself. It’s made of two parts:
$$ 5^3 $$
In this example:
- The base is the big number on the bottom (5). This is the number we are multiplying.
- The index (or exponent) is the small number at the top (3). This tells us how many times to multiply the base by itself.
So, $$ 5^3 $$ just means $$ 5 \times 5 \times 5 $$, which equals 125.
Analogy: Think of the index as a "repeat" button. $$ 5^3 $$ means "take the number 5 and repeat-multiply it 3 times".
Quick Review Box
Base: The number being multiplied.
Index/Exponent: The number of times the base is multiplied by itself.
Example: In $$ 7^4 $$, 7 is the base and 4 is the index.
The Rulebook: Laws of Positive Integral Indices
To make working with indices easier, there are 5 main rules you need to know. Think of them as the official rulebook that always works!
1. The Multiplication Law
When you multiply terms with the same base, you ADD the indices.
The Rule: $$ a^p \times a^q = a^{p+q} $$
Step-by-step Example: Let's simplify $$ 2^2 \times 2^3 $$.
1. Write it out the long way: $$ (2 \times 2) \times (2 \times 2 \times 2) $$
2. Count the total number of 2s. There are 5 of them!
3. So, the answer is $$ 2^5 $$.
4. The shortcut: Just add the indices! $$ 2^{2+3} = 2^5 $$. It works!
Memory Aid: When you Multiply, you get More, so you Add!
2. The Division Law
When you divide terms with the same base, you SUBTRACT the indices.
The Rule: $$ \frac{a^p}{a^q} = a^{p-q} $$
Step-by-step Example: Let's simplify $$ \frac{4^5}{4^2} $$.
1. Write it out: $$ \frac{4 \times 4 \times 4 \times 4 \times 4}{4 \times 4} $$
2. Cancel out the pairs of 4s from the top and bottom. You are left with three 4s on top.
3. The answer is $$ 4 \times 4 \times 4 = 4^3 $$.
4. The shortcut: Just subtract the indices! $$ 4^{5-2} = 4^3 $$. Perfect!
Common Mistake Alert! Make sure you subtract the bottom index from the top one. Don't divide the indices (e.g. 5 ÷ 2). You must subtract!
3. The Power of a Power Law
When you raise a power to another power, you MULTIPLY the indices.
The Rule: $$ (a^p)^q = a^{pq} $$
Step-by-step Example: Let's simplify $$ (3^2)^4 $$.
1. This means $$ 3^2 $$ multiplied by itself 4 times: $$ 3^2 \times 3^2 \times 3^2 \times 3^2 $$
2. Using the Multiplication Law, we add the indices: $$ 3^{2+2+2+2} = 3^8 $$.
3. The shortcut: Just multiply the indices! $$ 3^{2 \times 4} = 3^8 $$. So much faster!
4. The Power of a Product Law
When a product of different bases is raised to a power, the power applies to each base inside the bracket.
The Rule: $$ (ab)^p = a^p b^p $$
Example: Simplify $$ (2y)^3 $$.
The power of 3 applies to the 2 AND the y.
So, $$ (2y)^3 = 2^3 \times y^3 = 8y^3 $$.
5. The Power of a Quotient Law
When a fraction (or quotient) is raised to a power, the power applies to both the numerator and the denominator.
The Rule: $$ (\frac{a}{b})^p = \frac{a^p}{b^p} $$
Example: Simplify $$ (\frac{x}{5})^2 $$.
The power of 2 applies to the x AND the 5.
So, $$ (\frac{x}{5})^2 = \frac{x^2}{5^2} = \frac{x^2}{25} $$.
Key Takeaway: The 5 Main Laws
- Multiply Same Base: Add indices ($$ a^p \times a^q = a^{p+q} $$)
- Divide Same Base: Subtract indices ($$ \frac{a^p}{a^q} = a^{p-q} $$)
- Power of a Power: Multiply indices ($$ (a^p)^q = a^{pq} $$)
- Power of a Product: Apply power to each factor ($$ (ab)^p = a^p b^p $$)
- Power of a Quotient: Apply power to top and bottom ($$ (\frac{a}{b})^p = \frac{a^p}{b^p} $$)
Going Further: Zero and Negative Indices
Ready for something cool? The rules of indices lead to some interesting results when we get indices that aren't positive. It's all perfectly logical, so let's take a look.
The Zero Index
What happens if an index is 0? Let's use the Division Law to find out.
We know that $$ \frac{5^3}{5^3} = \frac{125}{125} = 1 $$.
But using the Division Law, $$ \frac{5^3}{5^3} = 5^{3-3} = 5^0 $$.
Since both are equal to $$ \frac{5^3}{5^3} $$, they must be equal to each other!
The Rule: $$ a^0 = 1 $$
This is true for any base 'a' (except for 0, which is a special case you don't need to worry about now). So, $$ 100^0 = 1 $$, $$ x^0 = 1 $$, and $$ (banana)^0 = 1 $$!
Negative Indices
A negative index might look scary, but it just means "flip it"! It tells you to take the reciprocal.
The Rule: $$ a^{-p} = \frac{1}{a^p} $$
How it works: Think about this pattern.
$$ 2^3 = 8 $$
$$ 2^2 = 4 $$ (divide by 2)
$$ 2^1 = 2 $$ (divide by 2)
$$ 2^0 = 1 $$ (divide by 2)
What comes next if we divide by 2 again?
$$ 2^{-1} = \frac{1}{2} = \frac{1}{2^1} $$
$$ 2^{-2} = \frac{1}{4} = \frac{1}{2^2} $$
Analogy: A negative sign in the index is like a ticket to move across the fraction line. If it's on top, it moves to the bottom and becomes positive. If it's on the bottom, it moves to the top!
Common Mistake Alert! A negative index does NOT make the number negative. For example, $$ 3^{-2} $$ is $$ \frac{1}{3^2} = \frac{1}{9} $$, it is NOT -9.
Key Takeaway: Zero and Negative Indices
- Zero Index: Anything to the power of 0 is 1. ($$ a^0 = 1 $$)
- Negative Index: It means "the reciprocal of". Flip it! ($$ a^{-p} = \frac{1}{a^p} $$)
- Great news! All 5 of the original laws still work perfectly with zero and negative indices!
Scientific Notation: For Really Big and Small Numbers
Ever tried to write down the distance from the Earth to the Sun? It's about 150,000,000,000 metres. That's a lot of zeros! Scientists use scientific notation to write these numbers more easily.
The Format
A number in scientific notation is written as:
$$ A \times 10^n $$
Where 'A' is a number between 1 and 10 (it can be 1, but not 10), and 'n' is an integer (a positive or negative whole number).
How to Convert to Scientific Notation
For BIG numbers (like 150,000,000,000):
- Find the decimal point (if you don't see one, it's at the end).
- Move the decimal point to the left until there is only ONE non-zero digit in front of it.
1.50000000000 - The number of places you moved the decimal is your positive index 'n'.
We moved it 11 places. - Write it in the correct format: $$ 1.5 \times 10^{11} $$
For small numbers (like 0.000025):
- Find the decimal point.
- Move the decimal point to the right until it's just after the first non-zero digit.
00002.5 - The number of places you moved is your negative index 'n'.
We moved it 5 places. - Write it in the correct format: $$ 2.5 \times 10^{-5} $$
Did you know?
The mass of an electron is about $$ 9.11 \times 10^{-31} $$ kg. Imagine writing that out without scientific notation! It would start with "0." followed by 30 zeros before you even get to the 9. That's why scientific notation is so useful!
Key Takeaway: Scientific Notation
- It's a shorthand for very big or very small numbers.
- Format: $$ A \times 10^n $$ (where $$ 1 \le A < 10 $$).
- Big number = Move decimal left = Positive index.
- Small number = Move decimal right = Negative index.
Number Systems: Binary and Denary
We count using ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). This is called the Denary system or Base-10. Computers are different. They only use two digits: 0 and 1. This is called the Binary system or Base-2.
Converting from Binary to Denary (Base-2 to Base-10)
To convert a binary number like $$ 1101_2 $$ to our normal denary system, we use place values that are powers of 2.
Step-by-step example: Convert $$ 1101_2 $$ to denary.
- Write down the binary number: 1 1 0 1
- Underneath it, write the place values, starting from the right with $$ 2^0=1 $$.
Binary Digit: 1 1 0 1
Place Value: $$ 2^3=8 $$ $$ 2^2=4 $$ $$ 2^1=2 $$ $$ 2^0=1 $$ - Multiply each binary digit by its place value:
$$ (1 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1) $$ - Add them all up: $$ 8 + 4 + 0 + 1 = 13 $$
So, $$ 1101_2 = 13_{10} $$.
Converting from Denary to Binary (Base-10 to Base-2)
To convert a denary number like $$ 13_{10} $$ to binary, we use repeated division by 2.
Step-by-step example: Convert $$ 13_{10} $$ to binary.
- Divide the number by 2 and write down the remainder.
- Keep dividing the result (the quotient) by 2 until the result is 0.
- Read the remainders from the bottom up to get your binary number.
13 ÷ 2 = 6 Remainder 1
6 ÷ 2 = 3 Remainder 0
3 ÷ 2 = 1 Remainder 1
1 ÷ 2 = 0 Remainder 1
Reading the remainders from the bottom up, we get 1101.
So, $$ 13_{10} = 1101_2 $$.
Key Takeaway: Number Systems
- Denary (Base-10): Our everyday system using digits 0-9.
- Binary (Base-2): The computer system using digits 0 and 1.
- To convert from Binary to Denary, multiply by place values (powers of 2).
- To convert from Denary to Binary, use repeated division by 2 and read remainders up.
Extra Challenge: The Hexadecimal System
This is an enrichment topic, so it's for those of you who want to explore a bit more!
Besides binary, another system used in computing is Hexadecimal, or Base-16. Since it's base-16, it needs 16 different symbols. It uses our normal digits 0-9, but then it needs six more! So, it borrows letters from the alphabet.
The symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Where:
- A = 10
- B = 11
- C = 12
- D = 13
- E = 14
- F = 15
You see this system used in things like colour codes on websites. For example, the code for pure white is #FFFFFF. This is a hexadecimal number!
Final Summary
Wow, you've learned a lot! You've mastered the 5 fundamental laws of indices, uncovered the mystery of zero and negative powers, learned the scientific way to write huge numbers, and even peeked into the language of computers with binary.
These concepts are like building blocks in mathematics. The more you practice using them, the easier they will become. Keep reviewing the rules and trying out problems. You've got this!