## Powers

## Writing mathematical power (or exponents)

Three squared is written 3^{2} = 3*3 ie 3 times itself twice. In the expression we just wrote the number 2 is written as a superscript number, ie a small number 2 to the right and above of the number 3 - this is the most common way a power expression is written.

Sometimes the symbol ^ is used as a way of writing powers, for example 3^2 meaning three squared. This is used where using superscript is too difficult, for example in computer code comments where a quick informal way of writing the maths is required.

We saw that some special powers have names - power of 2 is called squared. Power of 3 has a special name - cubed - and means multiplying something by itself three times - for example 5^{3} = 5*5*5. Powers are sometimes referred to using a name like 2nd- or 3rd- power meaning something square or cubed respectively.

Write four to the power three and its result in a way similar to this - 2^{2} = 4:

(1 mark)

Write 5^{3} in words (like two squared).

(1 mark)

What is 3^{2} times 3^{3?} Write your working:

(2 marks)

What is 243 expressed as a power of 3, ie 243 = 3^{what}

(2 marks)

Do you notice anything about the relationship between the two answers above? Explain.

(3 marks)

What is (4^{2)2} - ie (four squared) all squared? (Clue: do the calculation in the brackets first)

(2 marks)

What is 256 expressed as a power of 4 - ie 4^{what} = 256?

(2 marks)

Do you notice anything about the answers to the two last questions? Explain.

(3 marks)

What is the square-root of 25?

(1 mark)

In maths we can write square-roots using a special symbol - √16 = 4. But there's also another way to represent this which generalises the way to write roots - √25 = 5 = 25^{1/2}, ie twenty-five to the power of a half.

How might cube-root of 27 be written using power notation - 27^{1/what}

(2 marks)

Another way we write powers is using a negative value. An example of this is 4^{-2} = 1/(4^{2)} = what?

(1 mark)

The negative power means the sum is actually one divided by the same expression but using a positive power (see previous example).

Write out twenty-seven to the power of minus three but breaking it up into three parts - 27^{...} = 1/27^{...} = what

(2 marks)

Negative powers can also be combined with fractional powers.

What do you think 4^{-1/2} is?

(4 marks)

## Exponent

The word exponent is a more technical mathematical word for the power. The exponent in the expression 3^{2} = 9 is the number two.

What is the exponent in the expression 4^{3} = 64?

(1 mark)

## Some mathematical rules regarding exponents

### Glossary:

- Product - a fancy maths word for multiply - example - the product of 4 and 3 is 12.
- Quotient - a fancy maths word for dividing. Example the quotient of dividing twelve by three is four.

### Product rule:

3^{a} * 3^{b} = 3^{a+b}

### Power rule:

(3^{a)b} = 3^{a\}b)

### Quotient rule:

(3^{a} / (3^{b)} = 3^{a-b}.

Write three examples using the rules above to verify that they are correct:

(9 marks)

Here's an example of a sum where we expect a squared and square root to cancel each other out:

√(4^{2)} = (4^{2)1/2} = what

(2 marks)

Why do you get this answer (clue - use one of the rules earlier).

(3 marks)

What is (4^{2)} * (4^{-2),} ie times a number with positive power and negative power? Again use one of the rules earlier to explain this:

(3 marks)