Introduction to Square Roots
Math is radical!
—Bumper Sticker
This lesson exposes the meaning behind the sneaky √ symbol. What are roots and radicals? How can you estimate them?
Taking the square root (also called a radical) is the way to undo the exponent from an equation like 2^{2} = 4. If 2^{2} = 4, then 2 is the square root of 4 and √4 = 2. The exponent in 2^{2} tells you to square 2. You multiply 2 · 2 and get 2^{2} = 4.
The radical sign √ indicates that you are to find the square root of the number beneath it. The number inside the radical sign is called the radicand. For example, in √9, the radicand is 9.
A positive square root of a number is called the principal square root. For √9, 3 × 3 = 9, so 3 is the principal square root.
A negative sign outside the radical sign means the negative square root. For example, √9 = 3, but –√9 = –3.
Perfect Squares
The easiest radicands to deal with are perfect squares. A perfect square is a number with an integer for a square root. For example, 25 is a perfect square because its square root is 5, but 24 is not a perfect square because it doesn't have an integer for a square root. (No number multiplied by itself equals 24.)
Because they appear so often, it is useful to learn to recognize the first few perfect squares:
 0^{2} = 0
 1^{2} = 1
 2^{2} = 4
 3^{2} = 9
 4^{2} = 16
 5^{2} = 25
 6^{2} = 36
 7^{2} = 49
 8^{2} = 64
 9^{2} = 81
 10^{2} = 100
 11^{2} = 121
 12^{2} = 144
It is even easier to recognize when a variable is a perfect square because the exponent is even. For example, x ^{14} = x ^{7} · x ^{7} and a ^{8} = a ^{4} · a ^{4}.
Tip:You can use known perfect squares to estimate other square roots. Suppose you were asked to determine √50. What is the closest perfect square to 50? Try a few numbers to see:
So, you can determine that √50 is going to be between 7 and 8, much closer to 7. 
To determine if a radicand contains any factors that are perfect squares, factor the radicand completely. All the factors must be prime. A number is prime if its only factors are 1 and the number itself. A prime number cannot be factored any further. Let's try determining the square root of the following:
√ 64x 2y10
Write the number under the radical sign as a square:
√8xy5 · 8xy5
Because you have two identical terms multiplied by each other, you know this is a perfect square. Evaluate to find the final solution: 8xy^{5}. You could also have split the radical into parts and evaluated them separately.
Let's try another one. Find the square root of √ 64 x2 y 10.
First, split the terms:
√ 64 · x2 · y 10
Each term is a perfect square. Write as squares:
√8 · 8 · √x · x · √y5 · y5
Finally, evaluate each new radical:
8 · x· y^{5}
8xy^{5}

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