# Using index laws

**Multiplying terms**

Simplify the following:

*x* + *x* + *x* + *x* + *x + 5 *= 6*x*

Simplify the following:

*x* x *x* x *x* =* x³ *(or x to the power of 3, this is written using an index notation)

Point to note:

*x*x*x*x*x*cannot be written as*xxx*

**Multiplying terms involving indices**

Index notations can be used to simplify expressions.

For example,

4*p* x 2*p* = 4 x *p* x 2 x *p* = 8*p²*

*p* x *p²* = *p* x *p* x *p* = *p³*

*3p* x *p²* = 3 x *p* x *p* x *p* = 3*p³*

*3p* x 3*p* = (3*p)² = 9**p²*

Points to note:

- numbers together should be multiplied first then by letters in alphabetical order
- brackets are worked out before indices
- Indices are worked out before multiplication

**Multiplying terms with the same base**

When two terms are **multiplied** together with the **same base** the indices are **added, t**herefore:

For example,

a³ × a² = (a × a × a) x (a × a)

= a × a × a × a × a

= a⁵

= a(³+²)

**Dividing terms**

Remember, in algebra we do not usually use the division sign,

The division sign, ÷ is not usually used in algebra, therefore when dividing a number or a term, it is written like a fraction:

For example,

Similar to a fraction, expressions can be simplified by cancelling.

For example,

**Dividing terms with the same base**

When two terms are **divided** with the **same base** the indices are **subtracted**.

For example,

In general,

**Expressions of the form **

In algebra, there maybe terms that can be raised to a power and the result is that it is raised to another power.

For example,

Point to note:

- When a term is raised to a power and the result raised to another power, the powers are multiplied.

For example,

**The zero index**

When an expression has a zero indices the result is 1.

For example:

**Negative indices**

The rule of negative indices is as follows:

For example:

**Fractional indices**

The rules for fractional indices is as follows:

Also,