Squares and Square Roots



Squares and Square Roots

First learn about Squares, then Square Roots are easy.

How to Square A Number

To square a number, just multiply it by itself ...

Example: What is 3 squared?

|3 Squared |= |[pic] |= 3 × 3 = 9 |

"Squared" is often written as a little 2 like this:

[pic]

This says "4 Squared equals 16"

(the little 2 says the number appears twice in multiplying)

Squares From 12 to 62

|1 Squared |= |12 |= |1 × 1 |= |1 |

|2 Squared |= |22 |= |2 × 2 |= |4 |

|3 Squared |= |32 |= |3 × 3 |= |9 |

|4 Squared |= |42 |= |4 × 4 |= |16 |

|5 Squared |= |52 |= |5 × 5 |= |25 |

|6 Squared |= |62 |= |6 × 6 |= |36 |

|You can also find the squares on the | |[pic] |

|Multiplication Table: | | |

Negative Numbers

You can also square negative numbers.

Example: What happens when you square (-5) ?

Answer:

(-5) × (-5) = 25

(because a negative times a negative gives a positive)

That was interesting!

When you square a negative number you get a positive result.

Just the same as if you had squared a positive number:

[pic]

(For more detail read Squares and Square Roots in Algebra)

Note: if someone says "minus 5 squared" do you:

• Square the 5, then do the minus?

• Or do you square (-5) ?

You get different answers:

|Square 5, then do the minus: | |Square (-5): |

|-(5×5) = -25 | |(-5)×(-5) = +25 |

Always make it clear what you mean, and that is what the "( )" are for.

Square Roots

A square root goes the other way:

[pic]

3 squared is 9, so a square root of 9 is 3

A square root of a number is ...

... a value that can be multiplied by itself to give the original number.

A square root of 9 is ...

... 3, because when 3 is multiplied by itself you get 9.

It is like asking:

What can I multiply by itself to get this?

|[pic] |To help you remember think of the root of a tree: |

| |"I know the tree, but what is the root that produced it?" |

| |In this case the tree is "9", and the root is "3". |

Here are some more squares and square roots:

|[pic] |

|4 | |16 |

|5 | |25 |

|6 | |36 |

The Square Root Symbol

|[pic] |This is the special symbol that means "square root", it is sort of like a tick, and actually started |

| |hundreds of years ago as a dot with a flick upwards. |

| | |

| |It is called the radical, and always makes math look important! |

You can use it like this:

[pic]

you would say "square root of 9 equals 3"

Example: What is √25?

Well, we just happen to know that 25 = 5 × 5, so if you multiply 5 by itself (5 × 5) you will get 25.

So the answer is:

√25 = 5

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares

The perfect squares are the squares of the whole numbers:

| |1 |2 |3 |4 |5 |6 |

Note: we write down "3 Cubed" as 33

(the little "3" means the number appears three times in multiplying)

Some More Cubes

|4 cubed |= |43 |= |4 × 4 × 4 |= |64 |

|5 cubed |= |53 |= |5 × 5 × 5 |= |125 |

|6 cubed |= |63 |= |6 × 6 × 6 |= |216 |

Cube Root

A cube root goes the other direction:

3 cubed is 27, so the cube root of 27 is 3

|3 |[pic] |27 |

The cube root of a number is ...

... the special value that when cubed gives the original number.

The cube root of 27 is ...

... 3, because when 3 is cubed you get 27.

|[pic] |Note: When you see "root" think |

| |"I know the tree, but what is the root that produced it?" |

| |In this case the tree is "27", and the cube root is "3". |

Here are some more cubes and cube roots:

|[pic] |

|4 | |64 |

|5 | |125 |

|6 | |216 |

Example: What is the Cube root of 125?

Well, we just happen to know that 125 = 5 × 5 × 5 (if you use 5 three times in a multiplication you will get 125) ...

... so the answer is 5

The Cube Root Symbol

|[pic] |This is the special symbol that means "cube root", it is the "radical" symbol (used for square roots) with a|

| |little three to mean cube root. |

You can use it like this: [pic](you would say "the cube root of 27 equals 3")

You Can Also Cube Negative Numbers

Have a look at this:

|If you cube 5 you get 125: | |5 × 5 × 5 = 125 |

| | | |

|If you cube -5 you get -125: | |-5 × -5 × -5 = -125 |

So the cube root of -125 is -5

Perfect Cubes

The Perfect Cubes are the cubes of the whole numbers:

|1 |2 |3 |4 |5 |6 |7 |8 |9 |10 |11 |12 |13 |14 |15 |etc | |Perfect Cubes: |1 |8 |27 |64 |125 |216 |343 |512 |729 |1000 |1331 |1728 |2197 |2744 |3375 |... | |It is easy to work out the cube root of a perfect cube, but it is really hard to work out other cube roots.

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