Square Roots and Cube Roots

Lesson: Square Roots

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Definitions:

1. Square of a number:

It is the product of a number multiplied by itself.
Examples:
  1. 4 is square of 2 because 4 = 2 × 2 or 4 = -2 × -2
  2. 9 is square of 3, as 9 = 3 × 3 or 9 = -3 × -3
  3. 0.81 is also a square, as 0.89 = 0.9 × 0.9 or 0.89 = -0.9 × -0.9
  4. p2 is a square, as p2 = p × p or p2 = -p × -p
  5. x1/4 is also a square, as x1/4 = x1/2 × x1/2  or x1/4 = -x1/2 × -x1/2
Note: The square of a number is a number raised to power 2

2. Perfect Square:

A natural number is a perfect square if it is the square of a natural number.
Examples:
9 is a perfect square as it is the square of 3.

3. How to Find Whether a Number is a Perfect Square:

Express the given number as a product of its prime factors. If the powers (exponents) of the prime factors are even, then the number is a perfect square.
Examples:
1. Is 144 a perfect square?
Solution:
First, resolve 144 into its prime factors.
144 = 16 × 9 = 24 × 32
In the prime factors 2 and 3 above, the powers 4 and 2 are even numbers. Therefore, 144 is a perfect square.
2. Is 1000 a perfect square?
Solution:
First, resolve 1000 into its prime factors.
1000 = 23 × 53
In the prime factors 2 and 5 above, the powers are 3 and 3, i.e. odd numbers and not even. Therefore, 1000 is not a perfect square.

4. Properties of Perfect Squares:

1. Perfect squares are not negative.
2. Perfect squares do not end in 2, 3, 7 and 8.
3. Perfect squares end in any of the following digits: 1, 4, 5, 6, 9, 0
4. Perfect squares of even numbers are even and of odd numbers, odd.
5. Perfect squares do not have odd number of zeroes counted from the unit’s place upto the digit other than 0 to the left. Eg, 1000 is not a perfect square as it has 3 zeroes from right to left. 100 is a P.S. as there are 2 zeroes from left inward.

5. Square Root:

The square root of a number n is that whose square is n. Square root is denoted by this symbol: 2√ or just as √
Note:
In 2√ the index 2 need not be displayed and instead only the symbol √ need be written to denote square root of a number. But other indices need to be shown. 3√ denotes cube root of a number in which the index 3 cannot be dropped to convey cube root.
Examples:
The square root of 9 is 3; of 9/16 is 3/4
Note: The square root symbol √ denotes only the positive root of the number. Therefore, √9 = 3, √(9/16) = 3/4

6. How to Find the Square Root of a Number:

  • Step 1: Resolve the perfect square number into its prime factors.
  • Step 2: Divide whatever powers are in the prime factors by 2.
  • Step 3: Write the product of the prime factors with the quotients as obtained in step 2 in the powers’ place.
Examples:
1. What is the square root of 400?
Solution:
From step 1 above, 400 = 24 × 52.
From step 2 above, divide the two powers 4 and 2 each by 2. The quotients are 2 and 1 respectively.
From step 3 above, write the product: 22 × 5 = 20. Therefore, √400 is 20.
2. What is the square root of 0.00001764?
Solution:
0.00001764 = 1764/100000000
1764 = 36 × 49 = 62 × 72
(Though 6 is not a prime, still it is not necessary to resolve it into its prime factors 2 and 3, as you can see for yourself below)
Now, 0.00001764 = 1764/100000000 = (62 × 72)/108
Therefore, √ (62 × 72)/108) = (6 × 7)/104 = 42/10000 = 0.0042
More Solved Examples on Square Roots:
1. What least factor will make 250 a perfect square?
Solution:
Resolve 250 into its prime factors: 250 = 2 × 125 = 2 × 53
On multiplying 2 × 53 with 2 × 5 (i.e. 10), the prime factors 2 and 5 will have even numbers (2 and 4) as their exponents,
i.e. 250 ×10 = 2500 = 22 × 54
Now, 22 × 54 is a perfect square. So, 10 has to be multiplied to 250 to make 250 a perfect square.
2. Find the least number divisible by 5, 8 and 12 which is also a perfect square?
Solution:
LCM of 5, 8 and 12 is: 240. Now, 240 = 24 × 3 × 5.
The prime factors 3 and 5 have power 1 each. If 240 is multiplied by 3 and 5, then in the right, the powers of the prime factors 3 and 5 will be 2, even powers, i.e.
240 × 3 × 5 = 24 × 3 × 5 × 3 × 5 = 24 × 32 × 52
So, 240 × 15 = 3600. 3600 is the least number divisible by 5, 8 and 12 which is also a perfect square.