Part 2: Cube Roots_L08_U02_C3 Math_2023_Ready for Review

Connections 3 Lesson 8: Square and Cube Roots Part 2: Cube Roots_L08_U02_C3 Math_2023_Ready for Review

Part 2: Perfect Cubes and Cube Roots
Complete Part 2 of your Lesson 8 Interactive Notes as you work through your lesson.

A cube is a three-dimensional shape, with six square-shaped faces.

Construct a cube using eight of your Math U See Manipulative blocks.

What do you notice about this cube? How many dimensions does the cube have?

All of the faces of a cube are squares, having four equal sides.
A cube has three equal dimensions: length, width, and height.

To find the volume of a cube, multiply the length, width, and height together.

Like an actual cube, a number that is a perfect cube is multiplied by itself three times.
Numbers that are cubed are represented with an exponent of 3.

Example
2 cubed is written as $2^{3}$
$2^{3} = 2 \times 2 \times 2$

Common Perfect Cubes
Just like with perfect squares, the more you work with cubes the better you will get to know them.
Complete the Perfect Cubes table in Part 2 of your Lesson 8 Interactive Notes.

Exponent	Perfect Cube
$1^{3}$
$2^{3}$
$3^{3}$
$4^{3}$
$5^{3}$
$6^{3}$
$7^{3}$
$8^{3}$
$9^{3}$
$10^{3}$
$11^{3}$
$12^{3}$

Cube Roots

A cube root is one of three identical factors multiplied together that results in a perfect cube. They are recorded using a radical sign with an index of 3.

To calculate a cube root you need to determine the factor that when multiplied by itself three times will produce the desired perfect cube.

Example
$\sqrt[3]{27}$ To calculate the cube root of 27, you need to determine the factor that is multiplied three times to produce 27.
$3 \times 3 \times 3 = 27 \sqrt[3]{27} = 3$
The cube root of 27 is 3.
Multiplying 3 by itself three times equals the perfect cube of 27.

Try it out!
Evaluate the cube roots below.

$1) \sqrt[3]{8} 2) \sqrt[3]{125} 3) \sqrt[3]{343} 4) \sqrt[3]{64}$

When you are ready, click the solution button below to compare your solution.

**Text below appears when solution button is clicked.

Compare your solutions to those listed below.

$1) \sqrt[3]{8} 2 \times 2 \times 2 = 8 2 2) \sqrt[3]{125} 5 \times 5 \times 5 = 125 5 3) \sqrt[3]{343} 7 \times 7 \times 7 = 343 7 4) \sqrt[3]{64} 4 \times 4 \times 4 = 64 4$

**End solution button text

Can a cube root be negative?

Break it down using the rules of negative numbers.
negative x negative = positive
positive x negative = negative

Think about it.
Multiply $- 3^{3}$
$(- 3) (- 3) (- 3) = (- 27)$

Unlike square roots, the radicand of a cube root can be negative.

Example
$\sqrt[3]{- 125}$ A perfect cube is negative if its root is negative.
$(- 5) (- 5) (- 5) = (- 125)$
$\sqrt[3]{- 125} = (- 5)$
The cube root of -125 is -5.

Evaluate the Expression $- \sqrt[3]{64}$ Just like you learned when working with square roots, a negative sign (-) in front of a radical symbol is asking for the inverse, or opposite, of the root.

Example
$- \sqrt[3]{64} 4 \times 4 \times 4 = 64$ The cube root of 64 is 4.
The inverse opposite, of 4 is -4.
This expression simplifies to -4.

Try it out!
Evaluate each expression.

$1) \sqrt[3]{- 216} 2) \sqrt[3]{- 512} 3) - \sqrt[3]{27} 4) - \sqrt[3]{1000}$

$1) \sqrt[3]{- 216} (- 6) (- 6) (- 6) = (- 216) - 6 2) \sqrt[3]{- 512} (- 8) (- 8) (- 8) = (- 512) - 8 3) - \sqrt[3]{27} (3) (3) (3) = 27 T h e i n v e r s e o f 3 i s - 3 . - 3 4) - \sqrt[3]{1000} (10) (10) (10) = 1000 T h e i n v e r s e o f 10 i s - 10 . - 10$
**End solution button text

**Instructor Notes – Text will appear when button is clicked

How to Support Students:

The following activities can be used to support students that need additional practice or enrichment on cube roots and perfect cubes.

Which one doesn’t belong?
- Present students with a set of numbers.
- Three of the numbers should be perfect cubes, one is not
- Students must identify the number that is not a perfect cube.
Matching
- Create a matching game. Students will match a perfect cube to its root.
BINGO
- Create BINGO cards with a mix of roots and perfect cubes.
- Students mark matches on their cards as they are called.
- The first student with BINGO, wins!

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