Exponents Cubes and Cube Roots
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Skills Application: Exponents: Cubes and Cube Roots
Step-by-Step: Exponents: Cubes and Cube Roots
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The way to determine the size of an empty space is by filling it with cubic units, this can be done by using physical cubes or by doing the math on paper. Car manufacturers measure the irregular shape in a trunk space by actually filling it with foam cubes, then count them one by one to determine the volume of the trunk


A cube is a 3-dimensional object having equal length, width and height. Small cubes stacked together make a larger cube. The term “cube root” means the side length of any cube; and the term “cubing” means multiplying the side length of a cube by itself 3 times.

Exponents: Cubes
and Cube Roots

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In mathematics, we use exponents as a way to abbreviate repeated multiplication. Example:

The exponent form of 6 x 6 x 6 is 63. There are two parts to exponent notation:

1.  the base and

2.  the exponent.

The base tells you what number is being multiplied and the exponent tells you how many times this number is used as a factor. In the above example the base is 6 and the exponent is 3.


When the exponent is 3 the result is called the cube. To cube a number, we use the number in a multiplication 3 times. For example, 2 cubed means 2×2×2 which is 8. We can write 2 cubed as 23, 5 cubed (53) would be 5×5×5 = 125. You also have an x3 button on your calculator to speed up cubing. Find it and use it. See the practice problems.


A cube root is exactly the opposite direction. For instance, the cube root of 8 is 2. The symbol used for cube root is:

\[\sqrt[3]{  }\]

The cube root of 8 can be written as:

= 2.

 Finding the Cube root

How do you find the cube root of any number? Using a calculator is the easiest way.

Find out where the

\[\sqrt[3]{  }\]

button is on your calculator and whether it is a second function (2ndF) or not.  Find it and use it. See the practice problems.

Get a $20-or-so dual-display scientific calculator, even if you can`t take it with you into the exam hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you can`t do without. Trying out different functions on a calculator will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. It is important to try alternative calculations or approaches, to sketch & label angles, tracking changes to units of measure, visualizing to cross-link the concepts of volume and numbers.


A Short Review

Cubing a number or a unit of measure means multiplying three bases by themselves.

53 = 5 · 5 · 5

m3 = m · m · m

Whole numbers, fractions, decimals can all be cubed.

Cube rooting a number means dividing by the same divisor twice to find the number that was cubed in the first place.

 = 2, because 8 ÷ 2 ÷ 2 = 2, and 2 was cubed to get 8

 = 10, because 1000 ÷ 10 ÷ 10 = 10, and 10 was cubed to get 1000

Whole numbers, fractions, decimals can all be cube rooted.

GetMath Worksheet Level1 novertspaac

Worksheet: Level 1

The operations used are clearly specified. Only one type of mathematical operation is used in a task.

Worksheet: Level 1 Sample Questions


Write exponential notation as repeated multiplication:

1.  10³ = ___ x ___ x ___

Further examples:

What multiplication is meant by:

63³ = ___ · ___

7.2³ = ___ · ___

Worksheet: Level 1 Answer Key


Write exponential notation as repeated multiplication:

1.  10³ =  10 x 10 x 10

Download Worksheet: Level 1

GetMath Worksheet Level2 novertspaac

Worksheet: Level 2

Tasks involve one or two types of mathematical operation. Few steps of calculation are required.

Worksheet: Level 2 Sample Questions


Determine the cube roots of the following numbers. Use a calculator.

  1. \[\sqrt[3]{64}\]
    = _____

Further examples:

Calculate the following:

0.104³ = _____

= _____

Worksheet: Level 2 Sample Answer Key


  1. \[\sqrt[3]{64}\]
      =  4  (because 4 × 4 × 4 = 64)

Download Worksheet: Level 2

GetMath Worksheet Level3 novertspaac

Worksheet: Level 3

Tasks require a combination of operations. Several steps of calculation are required.

Worksheet: Level 3 Sample Questions


Calculate the following:

10in³ - 2in³ = ______

Further examples:

Remember: Volume units of measure can be added, subtracted, multiplied and divided with each other.

Calculate the following:

3m³ + 2m³ =

3m³ – 2m³ =

3m³ · 2 =

3m³ ÷ 2 =

Worksheet: Level 3 Sample Answer Key


10in³ - 2in³ =  8in³

Download Worksheet: Level 3


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