How to Calculate with Surds

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Finding distances and lengths often involves a type of square root called a surd. In this lesson, we show some interesting examples of calculating with surds.

Calculating with Surds

Johan wants to go hang out with his friends, but he has to finish his homework on surds first. He just doesn't get it - how can numbers be irrational? And how can he solve a math problem with square roots in it? Well, it's not as hard as it looks, Johan. Let's take a look at surds by first exploring the meaning of the word ''rational.''

Irrational Numbers and Surds

In math, the word rational has a special meaning. The first five letters: r-a-t-i-o form the word ''ratio.'' Rational numbers are defined with a ratio.

A ratio is one thing divided by another thing. In this lesson, that thing is an integer. We are ready for a definition: if a number is the ratio of two integers, the number is a rational number . For example, 4 is a rational number because 4 is 4/1. How about 4.2? Yes, 4.2 is a rational number because 4.2 is 42/10. What about a number like 0.666…? Well, 0.666… is a rational number because 0.666… is 2/3.

If a number is not a rational number, it is an irrational number. Here are some irrational numbers:


There's no way to write these numbers as the ratio of two integers. These numbers have a non-repeating decimal part. Note that we often use 22/7 as an approximation of π but π itself does not have a repeating pattern. It cannot be written as a ratio of two integers, so it is an irrational number.

What's √9? It's 3, since 9 is a perfect square. When the root of an integer results in an integer, we have a perfect square. Also, 3 is 3/1 meaning 3 is a rational number. Thus, √9 is a rational number. Some square roots, however, are irrational. The √2 is a non-repeating decimal. It has no repeating pattern. √2 is an irrational number. It's time for that ''surd'' word, says Johan.

Irrational roots of integers are called surds. Some examples of surds:


We can add and subtract surds together when they are the same. Just like x + 4x = 5x, we have √3 + 4√3 = 5√3.

Products, Quotients and Square Roots

Here's a useful fact: the square root of a product = the product of the square roots.

For example,


Step 1: Factor numbers into a products with a perfect square

Example: 200 is 100 times 2 and 90 is 9 times 10.

Step 2: Simplify

Example: Write √(200) as 10√2 because √(100 times 2) is √(100) times √2. And √(100) is 10.

Example: Write √90 as 3√10 because √(90) is √9 times √(10). And √9 is 3.

Step 3: Simplify further

Example: Replace '10 times 3' with 30.

Step 4: Combine surds

Example: Replace √2 times √(10) with √(20)

Taken separately, neither √2 nor √(10) may be further simplified. But √2 times √(10) is √(2 times 10) which is √(20).

Step 6: Simplify

Example: Write √(20) as √4 times √5.

Step 7: Simplify further

Example: √4 is 2 and 30(2) is 60.

The final answer is 60√5.

Here's another important fact to learn: the square root of a quotient = the quotient of the square roots


Step 1: Rewrite the quotient

Example: Write √(360) divided by √(20) as √(360/20).

Step 2: Simplify

Example: Replace 360/20 with 18.

Step 3: Factor with perfect squares

Replace √(18) with √(9) times √(2).

Step 4: Simplify

Example: √(18) is √(9) times √(2). And √(9) = 3.

The answer 3√2.

Difference of Squares

Johan might see surds in difference of squares calculations. In general, the product (a + b)(a - b) = a2 -a b +a b - b2 = a2 - b2. We multiply a sum of terms times a difference of terms. Out of this comes the square of one term minus the square of the other term. If these terms are surds, a surd will be squared. And since a surd is the square root of an integer, when we square a surd, we undo the square root, getting the integer by itself. For example,


Step 1: Identify the terms

Example: The terms are 1 and √2.

Step 2: Square each of these terms

Example: 12 is 1 and (√2)2 is 2.

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