Perfect Numbers- Squares and Cubes

Essential Question:How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?

Square and Cube Roots

Radicals (also called roots) are directly related to exponents.

Roots and Radicals

Square and Cube Roots

The simplest types of radicals are square roots and cube roots.

Radicals beyond square roots and cube roots exist, but we will not explore them here.

Roots and Radicals

Square and Cube Roots

The rules for radicals that you will learn work for all radicals – not just square roots and cube roots.

Roots and Radicals

Square and Cube Roots

The symbol used to indicate a root is the radical symbol -

Roots and Radicals

Perfect Cubes and Cube Roots

square cube

Area Volume

Square and Cube Roots

What does square root mean?

What does cube root mean?

Roots and Radicals

Remember:

A number is a perfect square if you multiply the same number such as: 1 x 1 unit squares and form them into a square. The side length of the square you form is that number’s square root.

Perfect Cubes and Cube Roots

3 x 3 = 9

3

3

The square root of 9 is 3

Remember:

A number is a perfect cube if you multiply the same number such as: 1 x 1 x 1 unit cubes and form them into a cube. The side length of the cube you form is that number’s cube root.

1 23 4

5 67 8

Cubes

2

2

2

2 x 2 x 2 = 8

3 x 3 x 3 = 27

If a number is a perfect square, then you can find its exact square root.

Remember

A perfect square is simply a number (or expression) that can be written as the square [raised to 2nd power] of another number (or expression).

Remember…….

If a number is a perfect cube, then you can find its exact cube root.

A perfect cube is simply a number (or expression) that can be written as the cube [raised to 3rd power] of another number (or expression).

Examples:

3327

3

5

4

125

64

36216

3273

5

4

125

643

62163

The End until next week

Thinking ahead: What if they aren’t perfect?

Non Perfect Squares and CubesEssential Question: How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?

Non-Perfect SquaresHere is the list of perfect squares perfect squares from 1 to 256.

11

24

39

416

525

636

749

864

981

10100

11121

12144

13169

14196

15225

16256

Not every number is a perfect square. If they aren’t, we call them non-perfect squaresnon-perfect squares. To find the square root of a number that is not a perfect square, we use estimation with perfect squares.

Estimating Square Roots

To calculate the square root of a non-perfect square

1. Place the values of the adjacent perfect squares on a number line.

2. Interpolate between the points to estimate to the nearest tenth.

Approximating Square RootsYou need to estimate its value of

non-perfect squares by determining which two perfect

squares it falls in between.Example: 11 is a non-perfect square 11 falls between perfect squares 9 & 16 Therefore, is between and

Since, = 3 and = 4Then is between 3 and 4

Estimating Square Roots

Example: 27

25 3530

What are the perfect squares on each side of 27?

36

Estimating Square Roots

Example: 27

25 3530

27

5 6

Estimate 27 = 5-6

36

√55 

√23

√5 

√14

√44

Find the two consecutive numbers the following non-perfect square fall between.

SHOW WORK!andandandandand

Between 7 & 8 Between 4 & 5 Between 2 & 3 Between 3 & 4 Between 6 & 7

Answer the following problem SHOW WORK!

If a square bedroom has an area of 144 square feet, what is the length

of one wall?

12 feet

First find the length of a side. Then you can use the length of a side to find the perimeter, the length of the fence.

Check It Out: Example

You want to build a fence around a square garden that is 250 square feet. Calculate the length of one side of the garden and the total length of the fence, to the nearest tenth.