An index or a power is a convenient way of indicating that a number is multiplied by itself several times. So 9²= 9×9, whereas 9⁵ = 9x9x9x9x9. We can use indices with numbers and also with letters in algebra (what does x⁴ mean?)

Geometrically, x² tells us the area of a square with side length x, and y³ tells us the volume of a cube with side length x.

Exercise

Let’s complete exercise 1 on pages 328 to 329 of the core textbook:

The answers are below:

Roots

In many areas of mathematics we encounter the concept of inverse operations. The inverse operations for x² and x³ are √x and ∛x. What do these operations indicate geometrically?

Exercise

Let’s complete exercise 2 on pages 330 and 331 of the core textbook:

The answers are below:

Negative and Zero Indices

Let’s memorise the formula for negative and fractional indices and practice in a few concrete cases.

Exercise

Let’s complete exercise 3 from page 331 of the core textbook:

The answers are below:

Arithmetic with Indices

We need to memorise the rules for multiplying and dividing powers of the same number and for finding the power of a power. We will need to apply this with numbers and with algebra.

Exercise

Let’s complete exercise 4 and exercise 5 on pages 332 and 333 of the core textbook:

The answers are below: