Roots - Year 7 PDF Download

Introduction

• Square Numbers: These are the results obtained when a number is multiplied by itself. For instance, 4 multiplied by 4 equals 16.
• Cube Numbers: A cube number results from multiplying a number by itself three times. An example is 3 cubed, which equals 27.
• Powers and Indices: Powers are represented by small digits called indices. For example, 3 to the power of 2 is written as 3².
• Square Roots: The opposite operation of squaring a number is finding the square root. It may not always yield an integer, in which case a calculator's square root (√) function is used, usually rounded to one decimal place.
• Cube Roots: The inverse process of cubing a number is determining the cube root. This is essential to find the original number when it has been cubed.

Finding the square root

• Squaring a number involves multiplying a number by itself. For instance, 4² is equivalent to 4 x 4.
• The opposite operation of squaring a number is known as finding the square root. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 36 is 6 since 6² equals 36. Additionally, -6 is also a square root of 36, as (-6)² equals 36.
• When determining the square root of a number, there are two solutions: a positive and a negative root. These roots can be expressed individually or with the ± symbol. For instance, the square root of 16 yields two results: 4 and -4, which can be written as ±4.
• The symbol for denoting the square root is √. For example, √36 represents "the square root of 36," which equals 6. It's beneficial to memorize the first 12 square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144.
  
• This image shows the link between squaring and finding the square root of a number. 3 squared is 9 (because 3² = 3 x 3 = 9). The opposite of squaring a number is called finding the square root. The square root of 9 is 3

Examples

Example 1:

• √100 means ‘the square root of 100’. Find the square root of 100
  
• Find the square root of 100. 10² means 10 x 10. This equals 100. The square root of 100 is 10

Example 2: 

• Find both roots of 16 (a positive and negative solution).
  
• The square root of 16 is 4 (because 4² = 4 x 4 = 16). The square root of 16 is also -4 (because -4² = -4 x -4 = +16). This is usually written as ±4

Finding the Cube Root

• A cube number is a number that is the result of multiplying a number by itself three times. For instance, 4³ equals 4 x 4 x 4.
• The opposite operation of cubing a number is known as finding the cube root. This process is represented by the symbol "³√". It's important to note that the cube root of a number may not always be a whole number. In such cases, you can use the "³√" function on a calculator and round the result to one decimal place.
• Memorizing the initial cube numbers like 1, 8, 27, 64, 125, and 216 can be beneficial in mathematical computations.
  
• This image illustrates the relationship between cubing and finding the cube root of a number. For instance, 4 cubed equals 64 (4³ = 4 x 4 x 4 = 64). Finding the cube root of 64 results in 4.

Examples

Example 1:

• ³√27 means ‘the cube root of 27’. Find the cube root of 27
  
• 3³ = 3 x 3 x 3 which equals 27. The cube root of 27 is 3

Example 2:

• Find the cube root of 90. 90 is not a cube number. The cube root of 90 therefore is not a whole number.

• 4³ = 64 and 5³ = 125. Therefore ³√90 lies between 4 and 5. Use the ³√ button on a calculator to find the exact value.
  
• ³√90 = 4∙4814... Round this answer to 1 decimal place. A number line can help when rounding. 4∙4814 rounded to 1 decimal place is 4∙5. ³√90 = 4∙5 (to 1 decimal place).