Chapter 10: Laws of Integral Indices
Hello! Welcome to the world of Indices. This might sound like a complicated word, but it's really just a mathematical shortcut. In this chapter, we're going to learn how to write and work with very big and very small numbers in a super simple way. It's a skill used everywhere, from calculating the distance to planets to understanding the size of tiny atoms. Let's get started!
What Exactly are Indices?
Imagine you have to write `5 x 5 x 5 x 5`. It's a bit long, right? Indices give us a shorter way to write this. We can write it as:
$$ 5^4 $$Here's what the parts mean:
- The big number, 5, is called the base. It's the number we are multiplying.
- The small number, 4, is called the index (or exponent, or power). It tells us how many times to multiply the base by itself.
So, `$$5^4$$` is just a neat way of saying "multiply 5 by itself 4 times". We read it as "5 to the power of 4".
Example: What is $$2^5$$?
The base is 2, and the index is 5. So, we multiply 2 by itself 5 times.
`$$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$$`
Quick Review Box
Base: The number being multiplied.
Index (or Exponent): How many times the base is multiplied by itself.
Power: The entire expression, including the base and index (e.g., `$$5^4$$` is a power).
The Laws of Positive Indices (Our Super Rules!)
To make working with indices even easier, there are some handy rules you need to know. Don't worry if they seem tricky at first, we'll break each one down with examples!
Rule 1: The Multiplication Law
The Rule: When you multiply powers with the same base, you add the indices.
$$ a^p \times a^q = a^{p+q} $$Think of it like this:
Let's look at `$$3^2 \times 3^4$$`.
The long way is: `(3 \times 3) \times (3 \times 3 \times 3 \times 3)`. If you count them, there are six 3s being multiplied together. So the answer is `$$3^6$$`.
The shortcut using our rule is: `$$3^2 \times 3^4 = 3^{2+4} = 3^6$$`. See? Much faster!
Memory Aid: When you Multiply, you Add. Think 'MA'!
Rule 2: The Division Law
The Rule: When you divide powers with the same base, you subtract the indices.
$$ \frac{a^p}{a^q} = a^{p-q} $$Think of it like this:
Let's solve `$$\frac{7^5}{7^2}$$`.
The long way is: `$$\frac{7 \times 7 \times 7 \times 7 \times 7}{7 \times 7}$$`. We can cancel out two of the 7s from the top and bottom, leaving `$$7 \times 7 \times 7$$`, which is `$$7^3$$`.
The shortcut using our rule is: `$$\frac{7^5}{7^2} = 7^{5-2} = 7^3$$`.
Memory Aid: When you Divide, you Subtract. Think 'DS' like the gaming console!
Common Mistake Alert!
These first two rules only work when the bases are the same! You cannot use the rule on `$$5^2 \times 6^3$$` because the bases (5 and 6) are different.
Rule 3: The Power of a Power Law
The Rule: When you raise a power to another power, you multiply the indices.
$$ (a^p)^q = a^{pq} $$Think of it like this:
Let's solve `$$(2^3)^2$$`. This means we have `$$2^3$$` two times: `$$2^3 \times 2^3$$`.
Using our first rule (the Multiplication Law), we know this is `$$2^{3+3} = 2^6$$`.
The shortcut using our new rule is: `$$(2^3)^2 = 2^{3 \times 2} = 2^6$$`.
Rule 4: Power of a Product Law
The Rule: A power outside the brackets applies to everything inside the brackets.
$$ (ab)^p = a^p b^p $$Think of it like this:
Let's try `$$(2 \times 5)^3$$`. This means `$$(2 \times 5) \times (2 \times 5) \times (2 \times 5)$$`.
We can re-arrange this to `$$(2 \times 2 \times 2) \times (5 \times 5 \times 5)$$`, which is just `$$2^3 \times 5^3$$`.
Rule 5: Power of a Quotient Law
The Rule: Just like with multiplication, a power outside a fraction applies to the top and bottom numbers.
$$ (\frac{a}{b})^p = \frac{a^p}{b^p} $$Example: `$$(\frac{3}{4})^2 = \frac{3^2}{4^2} = \frac{9}{16}$$`
Key Takeaway for Positive Indices
- Multiplying (same base): Add the indices.
- Dividing (same base): Subtract the indices.
- Power of a power: Multiply the indices.
- Power of a product/quotient: Apply the index to every part inside.
Zero and Negative Indices
What happens if an index isn't a positive whole number? Let's find out!
The Zero Index
This is a simple but very important rule. Anything to the power of zero is 1. (The only exception is `$$0^0$$`, which is undefined, but you won't need to worry about that!)
$$ a^0 = 1 $$But why? Let's use the Division Law to see. What is `$$\frac{5^3}{5^3}$$`?
We know that any number divided by itself is 1. So the answer must be 1.
Now let's use the Division Law: `$$\frac{5^3}{5^3} = 5^{3-3} = 5^0$$`.
Since both answers must be correct, it means that `$$5^0 = 1$$`!
Negative Indices
A negative index can look scary, but it just means "flip it" or "find the reciprocal".
$$ a^{-n} = \frac{1}{a^n} $$A negative index tells you to move the power to the bottom of a fraction and make the index positive.
Example 1: What is $$2^{-3}$$?
The negative sign tells us to flip it. So, we write it as a fraction with 1 on top.
`$$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$`
Why does this work? Let's use the Division Law again on `$$\frac{3^2}{3^5}$$`. Using the rule: `$$3^{2-5} = 3^{-3}$$`. Doing it the long way: `$$\frac{3 \times 3}{3 \times 3 \times 3 \times 3 \times 3} = \frac{1}{3 \times 3 \times 3} = \frac{1}{3^3}$$`. So, `$$3^{-3}$$` must be the same as `$$\frac{1}{3^3}$$`!
Good News! All the "Super Rules" we learned for positive indices work for zero and negative indices too!
Key Takeaway for Zero & Negative Indices
- Zero Index: `$$a^0 = 1$$` (anything to the power of 0 is 1).
- Negative Index: `$$a^{-n} = \frac{1}{a^n}$$` (flip it and make the index positive).
Scientific Notation: For Giant and Tiny Numbers!
Scientists often work with huge numbers (like the distance to stars) or tiny numbers (like the width of a cell). Writing out all the zeros is a pain! Scientific notation is a way to write these numbers neatly.
The format is always:
$$ A \times 10^n $$Where 'A' is a number between 1 and 10 (it can be 1, but not 10), and 'n' is an integer index.
How to write BIG numbers in scientific notation
Let's use the number 42,800,000.
Step 1: Move the decimal point so there is only one non-zero digit in front of it. The decimal point starts at the end: 42800000.
We move it to get 4.28.
Step 2: Count how many places you moved the decimal point. We moved it 7 places to the left. This number is our index.
Step 3: Write the number in the correct format. Since it's a big number, the index is positive.
$$ 4.28 \times 10^7 $$How to write SMALL numbers in scientific notation
Let's use the number 0.00091.
Step 1: Move the decimal point to get a number between 1 and 10.
We move it to get 9.1.
Step 2: Count how many places you moved the decimal point. We moved it 4 places to the right.
Step 3: Write the number in the correct format. Because the original number was small (less than 1), the index is negative.
$$ 9.1 \times 10^{-4} $$Did you know?
The name of the search engine "Google" was inspired by the number googol, which is 1 followed by 100 zeros. In scientific notation, a googol is written as `$$1 \times 10^{100}$$`. That's the power of indices!
Key Takeaway for Scientific Notation
- A neat way to write very big or very small numbers.
- Format: `$$A \times 10^n$$` where `$$1 \le A < 10$$`.
- Big numbers (greater than 1) have a positive index.
- Small numbers (less than 1) have a negative index.