Surds

Surds  are expressions that contain a square root, cube root or other roots. They are roots of numbers that produce an irrational number as a result, with infinite decimals. Therefore, they are left in their root form to represent them more exactly. For example, \(\sqrt2, \sqrt 3, \sqrt5, \sqrt 6, \sqrt 7, 2\sqrt 2\).

Surds Surds

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    Remember that an irrational number is a type of number that cannot be represented as a fraction.

    Simplifying surds

    To simplify surds, you need to remember the square roots of perfect squares. If the number inside the root of a surd has a square number as a factor, then it can be simplified.

    \(\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt 4 \cdot \sqrt 3 = 2 \cdot \sqrt 3 = 2\sqrt3\)

    The steps to follow to simplify surds are:

    • Write the number inside the root as the multiplication of its factors. One of the factors should be a square number.

    \(\sqrt{12} = \sqrt{4 \cdot 3}\)

    • Split the factors into separate roots

    \(\sqrt{4 \cdot 3} = \sqrt 4 \cdot \sqrt 3\)

    • Simplify the terms

    \(\sqrt 4 \cdot \sqrt 3 = 2 \cdot \sqrt 3\)

    • Take out the multiplication symbol

    \(2 \cdot \sqrt 3 = 2\sqrt3\)

    What are the rules for using surds?

    When working with surds, you need to remember the following rules:

    • Multiplying surds: As long as the index of the roots is the same, you can multiply surds with different numbers inside the root by simply combining them into one root and multiplying the numbers inside the root. Likewise, you can split a root into separate roots using factors.

    \(\sqrt a \cdot \sqrt b = \sqrt{a \cdot b}\)

    \(\sqrt2 \cdot \sqrt 5 = \sqrt{2 \cdot 5} = \sqrt{10}\)

    • Dividing surds: Similarly, as long as the index of the roots is the same, you can divide surds with different numbers inside the root by combining them into one root and dividing the numbers inside the root.

    \(\frac{\sqrt a}{\sqrt b} = \sqrt{\frac{a}{b}} \)

    \(\frac{\sqrt {10}}{\sqrt 2} = \sqrt{\frac{10}{2}} = \sqrt5\)

    • Multiplying a square root by itself: If you multiply the square root of a number by itself, you should obtain the original value.

    \(\sqrt{a} \cdot \sqrt{a} = (\sqrt{a})^2 = a\)

    \(\sqrt{3} \cdot \sqrt{3} = (\sqrt{3})^2 = 3\)

    • Multiplying a number by a surd: When multiplying a number by a surd, the order of the factors does not matter, and the result should be the number followed by the surd.

    \(a \cdot \sqrt{b} = \sqrt{b} \cdot a = a\sqrt{b}\)

    \(3\cdot \sqrt{2} = \sqrt{2} \cdot 3 = 3\sqrt{2}\)

    • Adding or subtracting surds: To add or subtract surds, the number inside the roots must be the same. You add or subtract the numbers outside the root.

    \(a\sqrt{x} + b\sqrt{x} = (a+b)\sqrt{x}\)

    \(a\sqrt{x} - b\sqrt{x} = (a-b)\sqrt{x}\)

    \(5\sqrt{3} + 2\sqrt{3} = (5+2)\sqrt{3} = 7\sqrt{3}\)

    \(5\sqrt{3} - 2\sqrt{3} = (5-2)\sqrt{3} = 3\sqrt{3}\)

    To add or subtract surds, you might need to simplify them first to find like terms.

    You cannot add \(\sqrt{2} + \sqrt{8}\), but you can simplify \(\sqrt{8}\) first,

    \(\sqrt 8 = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt2\)

    Then you can solve \(\sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}\)

    • Multiplying brackets containing surds: To multiply brackets containing surds, each term in the first bracket must be multiplied by each term in the second bracket. Then you can combine like terms.

    \(\begin{align} 2 +\sqrt{3})(5 +\sqrt{3}) &= 2 \cdot 5 + 2\sqrt{3} + 5 \sqrt{3} + (\sqrt3)^2 \\ &=10 +7\sqrt3+3 \\ &=13 +7\sqrt3 \end{align}\)

    Rationalizing the denominator of fractions containing surds

    The purpose of rationalizing the denominator of fractions containing surds is to eliminate the surd from the denominator. The strategy to do this is to multiply the numerator and denominator by the surd.

    • If the denominator contains only a surd:

    Rationalize the denominator in the following expression: \(\frac{5}{\sqrt3}\)

    Using the rules: \(a \cdot \sqrt{b} = \sqrt{b} \cdot a = a\sqrt b\) and \(\sqrt{a} \cdot \sqrt{a} = (\sqrt{a})^2 = a\)

    \(\frac{5}{\sqrt3} = \frac{5}{\sqrt3} \cdot \frac{\sqrt3}{\sqrt3} = \frac{5\sqrt3}{(\sqrt3)^2} = \frac{5\sqrt3}{3}\)

    • If the denominator contains a surd and a rational number: In this case, you need to multiply the numerator and denominator by the expression in the denominator, but with the sign in the middle changed, ie if it is (+) change it to ( -) and vice versa. This expression is called the conjugate.

    Rationalize the denominator in the following expression: \(\frac{(\sqrt6 + 3)}{(\sqrt6 -2)}\)

    The conjugate of \((\sqrt6 - 2)\) is\((\sqrt6 + 2)\)

    Multiplying the brackets and combining like terms, you can see that the surds in the denominator cancel each other.

    \(\frac{(\sqrt6 + 3)}{(\sqrt6 -2)} \cdot \frac{(\sqrt6 + 2)}{(\sqrt6 +2)} = \frac{6 +2\sqrt6+3\sqrt6 +6}{6 +\cancel{2\sqrt6 -2\sqrt6}-4} = \frac{12 +5\sqrt6}{2}\)

    Surds - Key takeaways

    • Surds are expressions that contain a square root, cube root or other roots, which produce an irrational number as a result, with infinite decimals. They are left in their root form to represent them more precisely.

    • To multiply and divide surds with different numbers inside the root, the index of the roots must be the same.

    • To add or subtract surds, the number inside the roots must be the same.

    • To add or subtract surds, they might need to be simplified first.

    • If the number inside the root of a surd has a square number as a factor, then it can be simplified.

    • The purpose of rationalizing the denominator of fractions containing surds is to eliminate the surd from the denominator.

    Frequently Asked Questions about Surds

    What is a surd?

    Surds are expressions that contain a square root, cube root or other roots, which produce an irrational number as a result, with infinite decimals. They are left in their root form to represent them exactly.

    What is an example of a surd?

    √2, √3, and 2√2 are examples of surds.

    What are the rules of surds?

    • Multiplying Surds: √a × √b = √(a × b)
    • Dividing Surds: √a/√b = √(a/b) = √(a÷b)
    • Multiplying the square root of a number 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

    • To multiply brackets containing surds, each term in the first bracket must be multiplied by each term in the second bracket.


    How do you add surds?

    To add or subtract surds, the number inside the roots must be the same.

    a√x + b√x = (a + b)√x

    How do you simplify surds?

    The steps to simplify surds are:

    • Write the number inside the root as the multiplication of its factors. One of the factors should be a square number 
    • Split the factors into separate roots 
    • Simplify the terms
    • Take out the multiplication symbol


    Test your knowledge with multiple choice flashcards

    What is the correct simplification of \(\sqrt{20}\)?

    Which of these is equivalent to \(7\sqrt{2}\)?

    How would you rationalize the fraction \[ \frac{4}{\sqrt{2}}?\]

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