# Year 9 Maths Chapter 6 - Indices and surds

## Index notation

``````3 x 3 x 3 x 3 x 3 x 3 x 3 = 3⁷
``````

## Scientific notation

science!

``````a x 10ᵐ
where m is an integer
and 1 <= a < 10
``````

## Prime factorisation

Writing a number as a product of its prime factors.

``````108 = 2 x 2 x 3 x 3 x 3
= 2² x 3³
``````

On Casio calculators, you can obtain the prime factorisation of any inputted number by pressing `SHIFT + DMS button`.

On Sharp calculators, well, I don’t know.

## Index laws

### Multiplication and division

``````aᵐ x bⁿ = aᵐ⁺ⁿ
aᵐ ÷ bⁿ = aᵐ⁻ⁿ
``````

### Powers and brackets

``````(aᵐ)ⁿ = aᵐⁿ
(ab)ᵐ = aᵐbᵐ
``````

### Powers and fractions

See Fractional indices for fractional indices.

``````(a/b)ᵐ = aᵐ/bᵐ
``````

### Zero index

``````a⁰ = 1
where a != 0
``````

### Negative indices

``````a⁻ᵐ = 1/aᵐ
``````

### Fractional indices

``````a¹/ᵐ = ᵐ√a
aˣ/ᵐ = ᵐ√aˣ
``````

## Surds

Surds are irrational numbers written with a radical sign. They cannot be expressed as a fraction.

For example, `5√6` is a surd. It means `5 x √6`.

``````  _
5√6
``````

## Surds

You can only add or subtract surds if they are like surds. For example, 5√6, 3√6 and √6 are like surds. If you were to add or subtract these together:

``````5√6 + 3√6 + √6 = 9√6
5√6 - 3√6 - √6 = 5√6
``````

### Multiplication and division

``````a√b x c√d = ab√cd

√a     / a \
–– = √| ––– |
√b     \ b /
``````

What a beautiful attempt at making a big bracket. I’m not bothered to go to the bracket making website right now so you’re going to have to deal with it.

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