Squares, Square Roots, Cube and Cube Roots

1.0 Lesson Objectives

• Understand the meaning and calculation of

  • 1.1 Square Numbers
  • 1.2 Square Roots
  • 1.3 Cube Numbers
  • 1.4 Cube Roots

We shall be starting our course with a relatively easy chapter. However, over the years, students have done multiple mistakes in these chapters that are easy to avoid if a few basic concepts are kept in mind. Thus, let us focus on each part in depth.

1.1 Square Numbers

The product of a number with itself is called the square of the number. In other words, square is the number that is formed when a number is MULTIPLIED with itself. There are a few things you must remember about square numbers:

• A square of a number can never be negative. It is always positive
  • For example the square of 2 is (2 x 2) = 4
  • The square of negative 2 or -2 is (-2 x -2) = 4
    • The two minus multiplied with each other form a positive.
    • You may recall that
      • Negative multiply with negative = positive
      • Positive multiply with positive = positive
      • Negative multiply with positive = negative
    • In other words, if two opposing signs are multiplied together, they end up in a negative result
    • A negative result is one with a minus sign, while a positive result is one with no sign or a plus sign.
• The question asked in the exam may ask you to take the square of a number by inserting a superscript.
  • Always remember, square is denoted by super script 2
    • For example if the examiner wants to ask you to take the square of 2, he may ask
      
    • You must remember that if there is ONLY a number, then the square involves its sign
      • For example
        is the same as (-2 x -2)
• A perfect square is one where the square is formed by multiplying the same integer with itself.  An integer is a number without decimals.
• 0 is not a perfect square. When a question asks you to add the first five perfect squares, you start from 1.

1.2 Square Roots

Square root is the opposite of square. It is used to find out which number was multiplied with itself to achieve the square that we have. A few crucial aspects to keep in mind are

• A square root is denoted by √
  sign.  Anything inside it is what you need to take the square root of. For example asks you to find the square root of 5.
• A square root ALWAYS has two answers, one positive and one negative. These are the positives and negatives of the same number.
  • For example, the square root of 25 is 5, however, it is not just positive 5. Instead, negative 5 or -5 is also a square root because -5 x -5 = 25 as well. Thus, our answer for square root of 25 is
    ±5 which represents positive and negative 5.
• At the O level stage, you must remember that the answer for square root of ANY NEGATIVE NUMBER is that there is no square root. Thus square root of -1, -100, -1000 etc. does not exist at this level of student.
• You must remember that square root is calculated using prime factors:
  • What is the square root of 10000
  • First try to find all the prime factors of the number. These are the prime numbers that multiply together to firm the number. Start from the smallest
    • Special aspect to remember, prime factors are numbers that are only divisible by themselves and 1. Also, these are always positive at this level. So the first prime number is 2, as 2 can only be divided completely by 1 and 2.
      • 1 is not a prime number because it does not have 2 divisors. It only has one divisor, ITSELF.
    • If we divide 10000 with 2 we get 5000
    • Again 5000/2 = 2500
    • 2500/2 = 1250
    • 1250/2 = 625
    • 625 is not divisible by 3 or 2, both of which are prime numbers. So we take the next prime number that is 5.
      • Special Tip – a number is only divisible by 3 is all the numbers in it add up to a number divisible by 3
        • So  1082582 = 1+0+8+2+5+8+2 = 26 which is not divisible by 3, so 1082582 is not divisible by 3
        • 129225 = 1+2+9+2+2+5 = 21 which is divisible by 3 ( 3 x 7 =21). So, 129225 is divisible by 3.
    • 625/5 = 125
    • 125/5 = 25
    • 25/5 = 5
    • 5/5 = 1
  • So we have 2, 2, 2, 2, 5, 5, 5, 5 as the prime factors. We can make two equal sets from it
    • 2 x 2 x 5 x 5 and 2 x 2 x 5 x5 are equal sets
    • Thus, our square root for 10000 is
      ±100
• At this stage, in O levels, you are sometimes asked to identify irrational numbers. Pro tip – square root of negative numbers are irrational.
• The first perfect square is 1.

1.3 Cube Numbers

Cube numbers are similar to square numbers. The main difference is that we multiply the same number with itself 3 times. It is denoted by a 3 in the super script

• Cube numbers can both be positive or negative. The cube of the same number in its positive and negative forms are different. For example the cube of 2 and -2 are NOT the same.
  • The cube of 2 or
    • 2 x 2 x 2
    • 8
  • The cube of -2
    • -2 x -2 x -2
    • Lets do it step wise
      • -2 x -2 = 4
      • 4 x -2 = -8
• The examiner may ask questions like evaluate
  
  • 10 x 10 x 10 = 1000

1.4 Cube Roots

Cube root is the same as square root, instead that we have to find the number that multiplies 3 times with itself to form the cube number. Thus, the cube root of 1000 is 10. However, it must be noted that there is only one sign, because cubes of same number with different signs have different signs as well.

• Cube root is denoted by
  
• The sign of the answer is extremely important in cube roots.
• The same prime factor method is used to determine cube roots as well.
• Find the cube root of 64
  • 64/2 = 32
  • 32/2 = 16
  • 16/2 =8
  • 8/2 = 4
  • 4/2 =2
  • 2/2 = 1
• We divided 6 times with 2, so we have 2, 2, 2, 2, 2, 2
• We need to to make 3 equal sets: 3 sets of 2 x 2 each.
• So the cube root is 2 x 2 = 4
• IF it was -64, the final answer would have a – as well. So it would have been -4.

1.1 Square Numbers

The product of a number with itself is called the square of the number. In other words, square is the number that is formed when a number is MULTIPLIED with itself. There are a few things you must remember about square numbers:

• A square of a number can never be negative. It is always positive
  • For example the square of 2 is (2 x 2) = 4
  • The square of negative 2 or -2 is (-2 x -2) = 4
    • The two minus multiplied with each other form a positive.
    • You may recall that
      • Negative multiply with negative = positive
      • Positive multiply with positive = positive
      • Negative multiply with positive = negative
    • In other words, if two opposing signs are multiplied together, they end up in a negative result
    • A negative result is one with a minus sign, while a positive result is one with no sign or a plus sign.
• The question asked in the exam may ask you to take the square of a number by inserting a superscript.
  • Always remember, square is denoted by super script 2
    • For example if the examiner wants to ask you to take the square of 2, he may ask
      
    • You must remember that if there is ONLY a number, then the square involves its sign
      • For example
        is the same as (-2 x -2)
• A perfect square is one where the square is formed by multiplying the same integer with itself.  An integer is a number without decimals.
• 0 is not a perfect square. When a question asks you to add the first five perfect squares, you start from 1.

1.2 Square Roots

Square root is the opposite of square. It is used to find out which number was multiplied with itself to achieve the square that we have. A few crucial aspects to keep in mind are

• A square root is denoted by √
  sign.  Anything inside it is what you need to take the square root of. For example asks you to find the square root of 5.
• A square root ALWAYS has two answers, one positive and one negative. These are the positives and negatives of the same number.
  • For example, the square root of 25 is 5, however, it is not just positive 5. Instead, negative 5 or -5 is also a square root because -5 x -5 = 25 as well. Thus, our answer for square root of 25 is
    ±5 which represents positive and negative 5.
• At the O level stage, you must remember that the answer for square root of ANY NEGATIVE NUMBER is that there is no square root. Thus square root of -1, -100, -1000 etc. does not exist at this level of student.
• You must remember that square root is calculated using prime factors:
  • What is the square root of 10000
  • First try to find all the prime factors of the number. These are the prime numbers that multiply together to firm the number. Start from the smallest
    • Special aspect to remember, prime factors are numbers that are only divisible by themselves and 1. Also, these are always positive at this level. So the first prime number is 2, as 2 can only be divided completely by 1 and 2.
      • 1 is not a prime number because it does not have 2 divisors. It only has one divisor, ITSELF.
    • If we divide 10000 with 2 we get 5000
    • Again 5000/2 = 2500
    • 2500/2 = 1250
    • 1250/2 = 625
    • 625 is not divisible by 3 or 2, both of which are prime numbers. So we take the next prime number that is 5.
      • Special Tip – a number is only divisible by 3 is all the numbers in it add up to a number divisible by 3
        • So  1082582 = 1+0+8+2+5+8+2 = 26 which is not divisible by 3, so 1082582 is not divisible by 3
        • 129225 = 1+2+9+2+2+5 = 21 which is divisible by 3 ( 3 x 7 =21). So, 129225 is divisible by 3.
    • 625/5 = 125
    • 125/5 = 25
    • 25/5 = 5
    • 5/5 = 1
  • So we have 2, 2, 2, 2, 5, 5, 5, 5 as the prime factors. We can make two equal sets from it
    • 2 x 2 x 5 x 5 and 2 x 2 x 5 x5 are equal sets
    • Thus, our square root for 10000 is
      ±100
• At this stage, in O levels, you are sometimes asked to identify irrational numbers. Pro tip – square root of negative numbers are irrational.
• The first perfect square is 1.

1.3 Cube Numbers

Cube numbers are similar to square numbers. The main difference is that we multiply the same number with itself 3 times. It is denoted by a 3 in the super script

• Cube numbers can both be positive or negative. The cube of the same number in its positive and negative forms are different. For example the cube of 2 and -2 are NOT the same.
  • The cube of 2 or
    • 2 x 2 x 2
    • 8
  • The cube of -2
    • -2 x -2 x -2
    • Lets do it step wise
      • -2 x -2 = 4
      • 4 x -2 = -8
• The examiner may ask questions like evaluate
  
  • 10 x 10 x 10 = 1000

1.4 Cube Roots

Cube root is the same as square root, instead that we have to find the number that multiplies 3 times with itself to form the cube number. Thus, the cube root of 1000 is 10. However, it must be noted that there is only one sign, because cubes of same number with different signs have different signs as well.

• Cube root is denoted by
  
• The sign of the answer is extremely important in cube roots.
• The same prime factor method is used to determine cube roots as well.
• Find the cube root of 64
  • 64/2 = 32
  • 32/2 = 16
  • 16/2 =8
  • 8/2 = 4
  • 4/2 =2
  • 2/2 = 1
• We divided 6 times with 2, so we have 2, 2, 2, 2, 2, 2
• We need to to make 3 equal sets: 3 sets of 2 x 2 each.
• So the cube root is 2 x 2 = 4
• IF it was -64, the final answer would have a – as well. So it would have been -4.

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