Coaching Centre
Surds and their arithmetic
What a surd is, the laws for multiplying and dividing square roots, simplifying surds by taking out square factors, and collecting like surds.
What is a surd?
A is a root that cannot be simplified to a rational number, such as $\sqrt{2}$ or $\sqrt{15}$. The symbol $\sqrt{x}$ always means the positive square root, so $\sqrt{9} = 3$, never $-3$. Cube roots can be negative: $\sqrt[3]{-8} = -2$, because $(-2)^3 = -8$.
Simplifying surds
Useful squares to know: $4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$.
Simplify $\sqrt{48}$ and $3\sqrt{50}$.
Adding and subtracting like surds
Like surds have the same number under the root. Add them like like terms: $5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}$. Often you must simplify first to see that surds are alike.
Simplify $\sqrt{75} - \sqrt{12} + \sqrt{27}$.
Simplify $2\sqrt{6} \times 5\sqrt{3}$.
Orange: y = √x. Dashed blue: y = k√x. Hover on the dashed curve: at x = 3 with k = 2 you get 3.46, which is √12. Taking a square factor out of the root is the same as stretching.
Simplify $\sqrt{98}$.
$98 = 49 \times 2$, so $\sqrt{98} = 7\sqrt{2}$.
$\sqrt{8} + \sqrt{18} =$
$2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$. You cannot add under the root.
Evaluate $\sqrt{8} \times \sqrt{18}$.
$\sqrt{144} = 12$.
$\sqrt{50} = k\sqrt{2}$. Find $k$.
$\sqrt{25 \times 2} = 5\sqrt{2}$.
$\sqrt{12} \times \sqrt{3} =$
$\sqrt{36} = 6$.
$\dfrac{\sqrt{45}}{\sqrt{5}} =$
$\sqrt{\tfrac{45}{5}} = \sqrt{9} = 3$.
$2\sqrt{27} - \sqrt{12} = k\sqrt{3}$. Find $k$.
$2 \times 3\sqrt{3} - 2\sqrt{3} = 4\sqrt{3}$.
Which of these is a surd?
The others equal $4$, $2$ and $\tfrac12$. $\sqrt{20} = 2\sqrt{5}$ is irrational.
Evaluate $\sqrt[3]{-27}$.
$(-3)^3 = -27$.
New numbers every time. Keep going until you get five in a row.