Square Roots Study Guide
Introduction to Square Roots
Math is radical!
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 22 = 4. If 22 = 4, then 2 is the square root of 4 and √4 = 2. The exponent in 22 tells you to square 2. You multiply 2 · 2 and get 22 = 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.
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:
- 02 = 0
- 12 = 1
- 22 = 4
- 32 = 9
- 42 = 16
- 52 = 25
- 62 = 36
- 72 = 49
- 82 = 64
- 92 = 81
- 102 = 100
- 112 = 121
- 122 = 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.
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: 8xy5. 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· y5