Powers

Writing mathematical power (or exponents)

Three squared is written 32 = 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 53 = 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 - 22 = 4:

(1 mark)

Write 53 in words (like two squared).

(1 mark)

What is 32 times 33? Write your working:

(2 marks)

 
 

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

(2 marks)

 
 

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

(3 marks)

 
 

What is (42)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 4what = 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 = 251/2, ie twenty-five to the power of a half.

How might cube-root of 27 be written using power notation - 271/what

(2 marks)

Another way we write powers is using a negative value. An example of this is 4-2 = 1/(42) = 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 32 = 9 is the number two.

What is the exponent in the expression 43 = 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:

3a * 3b = 3a+b

Power rule:

(3a)b = 3a\b)

Quotient rule:

(3a / (3b) = 3a-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:

√(42) = (42)1/2 = what

(2 marks)

 
 

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

(3 marks)

 
 
 

What is (42) * (4-2), ie times a number with positive power and negative power? Again use one of the rules earlier to explain this:

(3 marks)

 
 
 

Future: Logs and log tables?