Laws of Indices

Hi! Today we’re learning the laws of indices — also called the rules of powers. Don’t worry about the long name. These are simple shortcuts that save you from doing tons of multiplication. Once you learn them, problems that look scary become really easy. Let’s begin slow and build up step by step.

Step 1: What does “index” mean?

Let’s start with the basics. When you write something like 2⁵, you have two parts:

• The big number on the bottom (here, 2) is called the base
• The small floating number (here, 5) is called the index — also called the power or exponent

And what does 2⁵ actually mean? It just means “multiply 2 by itself, 5 times”:

So always remember: indices = repeated multiplication. Every rule below is just a smart way to count how many times the base appears.

Step 2: The 9 main rules

I’ll explain each rule with a simple example. Don’t try to memorise them all at once — read the why behind each one and they’ll naturally stick in your head.

Step 3: Negative powers — let’s clear up the confusion

This is the part students mess up most often, so I want to spend a little extra time on it. The biggest mistake I see is students writing things like:

2⁻³ = −8  ❌   (WRONG!)

Here’s the truth:

2⁻³ = 12³ = 18 = 0.125  ✓

A positive number with a negative power is still positive — it just becomes smaller, not negative. Look at this pattern:

• 2³ = 8   (big number)
• 2² = 4
• 2¹ = 2
• 2⁰ = 1   (the middle)
• 2⁻¹ = 12 = 0.5
• 2⁻² = 14 = 0.25
• 2⁻³ = 18 = 0.125   (small, but still positive!)

The numbers shrink as the power becomes more negative. They never turn negative themselves.

Step 4: Fractional powers — they’re just roots

Powers don’t always have to be whole numbers. Sometimes you’ll see fractions like 12, 23, or 54 in the power. When that happens, you’re being asked to take a root.

Here’s the full general rule that combines roots and powers:

In simple words: the bottom number is the root, the top number is the power. You can do either one first — the answer is the same.

Let me walk you through 8²³:

You could also do it the other way: 8² = 64, then ³√64 = 4. Same answer, but you’re working with bigger, scarier numbers. Always take the root first when possible — it keeps the numbers small and easier to handle.

What about negative AND fractional powers together?

No stress — just handle each thing one at a time. Negative means flip. Fraction means root.

Let’s try 16⁻³⁴:

Step 5: The “change the base” trick

Here’s a super useful idea that comes up all the time in IB exams.

The index laws ONLY work when the bases are the same. So if you see something like 2⁵ × 8³, you can’t just add the powers — because 2 and 8 are different bases.

BUT — wait! Notice that 8 is actually 2³. So if we rewrite 8 as 2³, suddenly both terms have the same base, and we can use our rules.

Three rules in three steps, and a messy expression became one clean power. That’s the magic of changing the base.

Step 6: Index laws with letters (algebra)

Everything we’ve learned works for algebra too. The base can be a letter (x, y, p, q…) instead of a number, and the rules don’t change one bit.

A few quick examples:

• x⁴ × x³ = x⁷   (same base, add powers)
• y⁹ ÷ y⁴ = y⁵   (same base, subtract)
• (z²)⁵ = z¹⁰   (power on power, multiply)
• (2p)⁴ = 16p⁴   (don’t forget the 2 also gets the power!)
• x⁻² = 1x²   (negative = flip)
• x¹² = √x   (fractional = root)

And when there are numbers AND letters together (like 3x²), treat them as separate jobs:

The 3 and 4 are coefficients — multiply them like normal numbers (3 × 4 = 12). The x² and x⁵ have the same base (x), so add the powers (2 + 5 = 7). Done.