Powers and roots: An explanation. 
First two years of secondary education.
 
Powers: base numbers and indices.

Luisa wants to know how many great-grandparents and great-great-grandparents she has had. She draws her family tree in order to count how many there are:

familia.gif (4016 bytes)

She has two parents (a mother and father).

Both her parents have 2 parents. Therefore, she has 2*2 = 4 grandparents.

Each of her grandparents had 2 parents, so she has 2*2*2 = 8 great-grandparents.

Each of her great-grandparents also had 2 parents; so she has 2*2*2*2 = 16 great-great-grandparents.

Operation Result
Parents 2 = 21 2
Grandparents 2*2 = 22 4
Great-grandparents 2*2*2 = 23 8
Great-great-grandparents 2*2*2*2 = 24 16

We often have to multiply a number by itself several times. Instead of writing 2*2*2*2 we can abbreviate it to 24  and we call this 'two to the power of four' which is a power.

24 is said as "2 raised to the power of 4" or "2 to the power of 4".

52 is said as "5 raised to the power of 2" or more commonly as "5 squared".

A power is the number of times a number is multiplied by itself. The number that is multiplied is called the base number and the number of times this number is multiplied is called the index or exponent.

In the number 24 the base number is 2 and the index is 4.

 

1. Calculate the following: 35,  53,   72,  27,  104,  410. Write down what the base number and index is in each case. 

Check your answers in the following window.

Some special powers.

2. Use the window above to work out the following:

  • 02,  05,  07,  010
  • 15,  18,  12,  110
  • 31,  51,  91,  101
  • 20,  30,  80,  100
  • 101,  102,  103,  104,   105,  106

Look carefully at your results and write a conclusion in your notebook for each of the five groups of numbers above.

Perfect squares.

Numbers raised to the power of 2 are called square numbers or perfect squares. We will be using lots of them in our maths lessons from now on.

3. Calculate the square numbers of the first 15 natural numbers and complete the following table in your notebook.

Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Square                              

Check your results in the following window.

Change the value of the base number to check your results.

As you know the area of a square whose side measures l is l2. Therefore, geometrically speaking, the square of a number is equal to the area of a square whose side is the same as the base number.

4.In the following window change the measurement of the side of the square to the first ten natural numbers and count how many little squares there are in each square formed.

Perfect cubes.

A Perfect cube or cube number is a number which is multiplied by itself three times.

5. Calculate the cubes of the first 15 natural numbers and complete the following table in your notebook.
Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Cube                              

Work out the answers in your notebook and check them in the following window.

Negative base numbers and powers.

Work out the following: (-5)3 and (-5)4.

(-5)3 = (-5)*(-5)*(-5) = -125. The answer is negative.

(-5)4 = (-5)*(-5)*(-5)*(-5) = 625. The answer is positive.

In general, if we raise a negative base number to an even power the result is always positive. If we raise a negative base number to an odd power the result is always negative.

7. Work out the following and check your results in the following window.

  1. (-3)5
  2. (-3)6
  3. (-4)4
  4. (-4)5
  5. (-10)5
  6. (-13)9
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  Fernando Arias Fernández-Pérez
 
Spanish Ministry of Education. Year 2001
 
 

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