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Simplifying Radicals Study Guide

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Updated on Aug 24, 2011

Find practice problems and solutions for these concepts at Simplifying Radicals Practice Problems.

This lesson defines radicals and shows you how to simplify them. You will also learn how to add, subtract, multiply, and divide radicals.

What Is a Radical?

You have seen how the addition in x + 5 = 11 can be undone by subtracting 5 from both sides of the equation. You have also seen how the multiplication in 3x = 21 can be undone by dividing both sides by 3.Taking the square root (also called a radical) is the way to undo the exponent from an equation like x2 = 25.

The exponent in 72 tells you to square 7. You multiply 7 · 7 and get 72= 49.

The radical sign  in √36 tells you to find the positive number whose square is 36. In other words, √36 asks: What number times itself is 36? The answer is √36 = 6 because 6 · 6 = 36.

The number inside the radical sign is called the radicand. For example, in √9, the radicand is 9.

Square Roots of Perfect Squares

The easiest radicands to deal with are perfect squares. 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, and 122= 144. It is even easier to recognize when a variable is a perfect square, because the exponent is even. For example: x14= x7· x7,or (x7)2, and a8= a4· a4, or (a4)2.

Example:64 x2 y10

Write as a square.                                             √8xy5 · 8xy5

Evaluate.                                                              8xy5

You could also have split the radical into parts and evaluated them separately:

Example:64 x2 y10  
Split into perfect squares. 64 · x2 · y10
Write as squares. 8 · 8 · √x · x · √y5 · y5
Evaluate. 8 · x · y5
Multiply together. 8xy5

If your radical has a coefficient like 3√25, evaluate the square root before multiplying: 3√25 = 3 · 5 = 15.

Simplifying Radicals

Not all radicands are perfect squares. There is no whole number that, multiplied by itself, equals 5.With a calculator, you can get a decimal that squares to very close to 5, but it won't come out exactly. The only precise way to represent the square root of 5 is to write √5. It cannot be simplified any further.

There are three rules for knowing when a radical cannot be simplified any further:

  1. The radicand contains no factor, other than 1, that is a perfect square.
  2. The radicand is not a fraction.
  3. The radical is not in the denominator of a fraction.
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