Squares And Perfect Squares

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What Is The Square Of A Number?

The square of a number means to multiply the number by itself.

Example:
The square of 3 is 3 × 3 = 9

The square of a number can be written in exponent notation such as 3² where 3 is base and 2 is the exponent.

3² is read as “three to the second power” or “three squared”.

What Are Perfect Squares?

Perfect squares are the squares of whole numbers.

Example:
1, 4, 9, 16, 25 and 36 are the first 6 perfect squares because
1² = 1 × 1 = 1
2² = 2 × 2 = 4
3² = 3 × 3 = 9
4² = 4 × 4 = 16
5² = 5 × 5 = 25
6² = 6 × 6 = 36

How To Check For Perfect Squares?

We use repeated division by prime factors to check whether a given number is a perfect square.

Example:
Check whether 441 is a perfect square.

Solution:

441 = 3 × 3 × 7 × 7
  = 3 × 7 × 3 × 7
  = 21 × 21
  = 21²

So, 441 is a perfect square.

How To Determine If A Number Is A Perfect Square?

We can use prime factorization to check if a number is a perfect square.

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How To Find Smallest Positive Whole Number That Is A Perfect Square And A Multiple Of A Specific Whole Number?

Examples:

1. What is the smallest positive whole number that is a perfect square and is a multiple of 24?
2. What is the smallest positive whole number that is a perfect square and is a multiple of 150?

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How To Find The Square Of Negative Numbers, Decimals And Fractions?

We can also have the square of negative numbers, decimals and fractions.

When calculating the square of a number, take note of the following:

1. The square of a number is always positive.
   Example:
   (−5)² = (−5) × (−5) = 25
   (−7)² = (−7) × (−7) = 49

Observe two important properties of a square in the last example above:
a) The square of a negative number becomes a positive number.
b) The square of a signed number is the same as the square of the unsigned number, i.e. (−7)² = 49 = 7².

2. The square of a decimal will have twice the number of decimal places as the original decimal.
   Example:
   (0.3)² = 0.3 × 0.3 = 0.09 ( 1 d.p. after squaring becomes 2 d.p.)
   (0.03)² = 0.03 × 0.03 = 0.0009 ( 2 d.p. after squaring become 4 d.p.)
   10.2² = 10.2 × 10.2 = 104.04 ( 1 d.p. after squaring becomes 2 d.p.)

3. To square a fraction, multiply the numerator by itself and do the same for the denominator.
   Example:
   
   
   

Take note that if a positive fraction which is less than 1 is squared, the result is always smaller than the original fraction.

4. To square a mixed number, change it to an improper fraction before calculating the square.
   Example:
   

Squaring Negative Numbers

Look for parentheses to group negative numbers that are to be squared.

Example:
Simplify (−3)² and −3²

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