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Understanding Surds

Surds are mathematical expressions involving square roots, cube roots, or other roots. They are numbers that result in an irrational value with infinite decimals and are typically represented in their root form for precision. For instance, both √2 and √3 are considered surds.

When dealing with surds, it's essential to keep in mind the following rules:

  • When adding or subtracting surds, they must have the same number inside the root to combine like terms.
  • When multiplying or dividing surds with different numbers inside the root, the indices of the roots must be equal.
  • Surds may need to be simplified before they can be added or subtracted.
  • If the number inside the root of a surd has a square number as a factor, it can be simplified.

Rationalizing the Denominator

To rationalize the denominator of fractions containing surds, the aim is to eliminate the surd from the denominator. This process involves following two rules:

  • If the denominator contains only a surd, it can be rationalized by multiplying the numerator and denominator by the expression in the denominator.
  • If the denominator contains a surd and a rational number, the numerator and denominator should be multiplied by the conjugate of the expression in the denominator, with the sign in the middle changed.


Let's rationalize the denominator of 1/(4+√x).

Solution: The expression in the denominator is 4+√x, so we need to multiply the numerator and denominator by its conjugate, which is 4-√x.

  • 1/(4+√x) × (4-√x)/(4-√x) = (4-√x)/(16-x)

The surds in the denominator have been eliminated, and the fraction is now simplified.

The Key Points to Remember

Here are the main takeaways about surds:

  • A surd is a mathematical expression containing roots that result in an irrational number.
  • To multiply or divide surds, the roots' indices must be the same.
  • To add or subtract surds, they must have the same number inside the root.
  • Rationalizing the denominator aims to eliminate the surd from the denominator.

Defining Surds

A surd is a mathematical expression that involves a root, resulting in an irrational number. They are usually represented in their root form, such as √2, √3, and 2√2.

Rules for Working with Surds

When working with surds, the following rules should be observed:

  • Multiplying Surds: √a × √b = √(a × b)
  • Dividing Surds: √a/√b = √(a/b) = √(a÷b)
  • Multiplying a Square Root by Itself: √a × √a = (√a)² = a
  • Multiplying a Number by a Surd: a × √b = √b × a = a√b
  • Adding or Subtracting Surds: a√x + b√x = (a + b)√x a√x - b√x = (a - b)√x

Simplifying Surds in a Few Steps

To simplify surds, follow these steps:

  • Represent the number inside the root as the multiplication of its factors.
  • Break the factors into separate roots.
  • Simplify the terms.
  • Remove the multiplication symbol.

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