# 4 Squares and other powers

Multiplying a number by itself is called **squaring** it and there is a key on scientific and graphics calculators which does this. On the TI-84 the key is marked .

## Exercise 4: Press some more

Try using the squaring key in the following key sequences. (If your calculator has an key then use it instead of ).

(a)

**3**(b)

**0.1**(c)

**4**(d)

**4**(e)

**3****2**(f)

**3****2**

Note any observations you have about the operation of the squaring key.

### Answer

On the TI-84 you should get (a) 9, (b) .01, (c) ^{−}16, (d) 16, (e) .75, (f) 2.25

Unless there are brackets, the function operates only on the preceding number (without any sign) so in **4** the 4 is squared before the is applied. If you want to square ^{−}4, you need brackets.

Another way of saying ‘five squared’ is ‘five to the power 2’ and it is reasonable to extend this to other powers:

5 × 5 is written as 5^{2} and is said ‘five to the power 2’

5 × 5 × 5 is written 5^{3} and is said ‘five to the power 3’

5 × 5 × 5 × 5 is written 5^{4} and is said ‘five to the power 4’ and so on.

There is a key on the calculator which can be used to carry out such calculations involving powers. It is usually marked as *y ^{x}* or ^. On the TI-84, it is the key, and this key is on the right-hand side of the keyboard, above the operation keys.

To work out 2 × 2 × 2 × 2 × 2 or 2^{5}, for example, you press **2** **5**. Similarly 0.6^{3} is the same as 0.6 × 0.6 × 0.6, which is 0.6^{3} or **0.6** **3**. Try these now on your own calculator. Does it matter if you enter 0.6 or just .6?

You should get 2 5 is 32 and 0.6 3 is the same as .6 3 – they are both .216.

## Brain stretcher: Negative powers

What possible meaning could there be for negative powers, such as 7^{−3}?

Try using the power key to work out the value of 2^{−1}.

Does the answer look familiar? Try 2^{−2}, 2^{−3}, 2^{−4} …

Can you predict what 10^{−1}, 10^{−2}, 10^{−3} … might mean? Check your prediction on the calculator.

## Brain stretcher: Fractional powers

Press the following:

**1 6** **1** **2**

What is going on here?

Press **[ENTRY]** and change the 16 to other numbers such as 20, 25…

Once you know the effect of raising to the power , try other fractions such as and .