## More Algebra

### Review

An index, power or exponent is a way of shortening expressions such as 4 × 4 × 4 × 4 × 4 to 45.

3p4 is a short way of writing 3 × p × p × p × p.

The 3 is called the coefficient , the p is called the base or variable and the 4 is called theindex, power or exponent.

Indices (plural of index) obey certain rules when they are being multiplied and divided.

### Multiplication of terms

When multiplying numbers or variables with indices, add the indices together.
The bases must be the same number or variable. The numbers are multiplied as normal.

Examples

 Long way Quick way Answer a2 × a4 (a × a) × (a × a × a × a) a(2 + 4) a6 2p3 × 4p5 2 × 4 × (p × p × p) × ( p × p × p × p × p ) 2 × 4 × p(3 + 5) 8p8

### Division of terms

When dividing numbers or variables with indices, subtract the indices.
The bases must be the same number or variable. The numbers are divided as normal.

Examples

 Long way Quick way Answer p5⁄p3 p × p × p × p × pp × p × p p(5 − 3) p2 8m6⁄4m2 8 × m × m × m × m × m × m4 × m × m (8 ÷ 4) × m(6 − 2) 2m4

### Raising a term with an index to to another index

If a term with an index is raised to another index then the indices should be multiplied.
Any numbers inside the bracket must be raised to the outside index as well.

Examples

 Long way Quick way Answer (q2)3 (q × q) × (q × q) × (q × q) q2 × 3 q6 (4z2)3 (4z2) × (4z2) × (4z2) 43 × z(2 × 3) 64z6

### Special Indices

There are two rules for "special" indices.

1. Any number or variable to the power of zero = 1

2. Any number or varaible to the power of one = itself

Examples

 30 1 p0 1 4q0 4 × 1 = 4 (3x2)0 1 51 5 6x1 6x