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Square Roots Study Guide

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Updated on Oct 3, 2011

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 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.

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:

    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.

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:

6 × 6 = 36
Too low.
8 × 8 = 64
Too high.
7 × 7 = 49
Perfect.

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

8xy5

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