Math is symmetrical and full of inverse operations. Squaring a number and finding the square root are opposites of each other. To understand squares and square roots, think of a square. Every side of a square is the same length. Multiplying two sides gives you the total number of interior units that make up the whole square. The square root of a number is the length of the side, while the square is the total number of units inside the square.
To square a number, multiply that number by itself. For example, 6 squared is 6 x 6 = 36. If you create a square with a side length of 6, you have 6 rows of 6. The total number of units in that square is 36.
You express 6 squared as 6^2. The 2 is the power, and it indicates the number of times you multiply a number by itself. You could also say that you are raising 6 to the second power. Squaring a number is unique because only multiplying a number by itself creates a perfect square. It is a common mistake to multiply 6 x 2 when you see 6^2, but 12 units does not create a perfect square.
Finding the Square Root
Finding the square root of a number means looking at the whole square to determine the side length, meaning the number that was multiplied by itself. A square is presented under a radical sign. Without a calculator that has a radical button, you must know your multiplication facts to determine the square root. To find the square root of 144, you must know that 12 x 12, or 12^2, is 144. Perfect squares never end in 2, 3 or 8; squares with these endings will be long, irrational decimals. When figuring manually, you must estimate these square roots instead.
Estimating Square Roots
Sometimes a square is not perfect, meaning the square root is not a whole number. For example, the square root of 11,548 ends in an 8, and if you plug it into a calculator, the square root is 107.4616210.... Without a calculator, you could look at 11,548 and see that it is between 100^2, which is 10,000, and 110^2, which is 12,100. Looking at the hundreds place, note that 548 pushes the number closer 12,100, so the square root will be closer to 110 than 100. The last two digits, 48, are closest to the perfect square 49, which has a square root of 7. So, you could estimate that the square root of 11,548 is around 107.
It is impossible to have a negative perfect square. A square comes from multiplying a number by itself. Following the rules for multiplication, the product of two negative numbers and the product of two positive numbers will always be positive, so you can only multiply two negative numbers or two positive numbers to get a perfect square. Any problem that asks for the square root of a negative number under the radical sign is a trick question and will have "no solution." You will usually see square root answers written with a plus-or-minus symbol. This means the original square root could have been either positive or negative because both give a positive square. If there is a negative symbol in front of the radical sign, that means that the square root will be negative.