Understanding Indices: Examples and Practice

Basic Index Rules You Need to Know

Learning index rules is like having a calculator built into your brain – they make tricky maths problems surprisingly simple! When you multiply numbers with the same base, you just add the powers together. For example, x² × x⁶ = x⁸ because 2 + 6 = 8.

Division works the opposite way – you subtract the powers instead. So y² ÷ y³ = y⁻¹ because 2 - 3 = -1. This might give you a negative power, but don't worry – that's perfectly normal!

Here's a handy fact: anything to the power of 0 equals 1. So m⁰ = 1, no matter what m is (as long as it's not zero). When you see brackets with powers inside and outside, like (k⁴)³, you multiply them together: 4 × 3 = 12, so the answer is k¹².

Quick Tip: Remember "MADSPM" – Multiply Add, Divide Subtract, Power Multiply!

Negative and Fractional Powers Made Simple

Negative powers aren't scary once you know the secret – they just mean "flip it and make it positive." So 3⁻² = 1/3² = 1/9. Think of the negative sign as instructions to put the number on the bottom of a fraction instead.

Fractional powers are actually roots in disguise. When you see 25^$1/2$, the bottom number (2) tells you it's a square root, so the answer is 5. For something like 9^$3/2$, you take the square root first (getting 3), then cube it (getting 27).

The trickiest bit is combining negative and fractional powers, like 16^$-3/2$. First work out 16^$3/2$: take the square root (4), then cube it (64). Because of the negative sign, your final answer is 1/64.

When dealing with fractions like $4/9$^$3/2$, apply the power to both the top and bottom separately. Square root of 4 is 2, cubed gives 8. Square root of 9 is 3, cubed gives 27. So your answer is 8/27.

Remember: With negative fractional powers of fractions, work out the positive version first, then flip your final answer!