A surd is a root that cannot be written exactly as a fraction. Working with them exactly, rather than rounding, is what keeps later answers precise.
Today's hook, Your calculator shows $\sqrt{2} = 1.414213562...$ and then stops. It has to, because the decimal never terminates or repeats. Surd form is how you write the exact value instead of a rounded one.
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Recall, your gut answer first
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Recall, your gut answer first
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Is $\sqrt{9}$ a surd? Is $\sqrt{10}$? Decide what makes a root a surd, then estimate $\sqrt{50}$ to the nearest whole number without a calculator.
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
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Simplifying a surd means extracting perfect squares
Work through the core explanation before applying it.
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Simplifying a surd means extracting perfect squares
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A surd is simplified when the number under the root sign has no perfect square factor left. Finding the largest perfect square factor is the whole technique.
$\sqrt{72}$ is not simplified, because $72 = 36 \times 2$ and 36 is a perfect square. So $\sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$. Using a smaller square such as 4 also works, it just takes more steps.
$\sqrt{ab} = \sqrt{a}\sqrt{b}$ is true. $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$ is false; test it with 9 and 16.
Only like surds add
$3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$, but $3\sqrt{5} + 2\sqrt{7}$ cannot be combined. Treat $\sqrt{5}$ like a variable.
Simplify before adding
$\sqrt{8} + \sqrt{18}$ looks unlike until you simplify: $2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
A surd is a root whose exact value is irrational, such as $\sqrt{2}$ or $\sqrt[3]{5}$.
$\sqrt{ab} = \sqrt{a}\sqrt{b}$ and $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$, but roots do not split over addition.
Like surds have the same number under the root sign and can be added or subtracted.
Brackets containing surds expand exactly as algebraic brackets do.
Understand
Concepts
Why $\sqrt{9}$ is not a surd while $\sqrt{10}$ is.
Why extracting the largest perfect square factor is the efficient route.
Why $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$.
Can do
Skills
Simplify a surd by extracting perfect square factors.
Multiply, divide, add and subtract surds.
Expand brackets containing surds, including squaring a binomial surd.
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Key terms
SurdA root that cannot be written exactly as a fraction, so its decimal never ends or repeats. Like this: $\sqrt{2}$ is a surd, but $\sqrt{9} = 3$ is not.
Rational numberA number that can be written as one whole number over another. Like this: 3, $\frac{7}{4}$ and $0.25$ are rational; $\sqrt{2}$ is not.
Perfect squareA whole number that is another whole number squared, used to simplify surds. Like this: 36 is a perfect square because $6^2 = 36$, so $\sqrt{72} = 6\sqrt{2}$.
Simplified surd formA surd where the number under the root has no perfect square factor left. Like this: $6\sqrt{2}$ is simplified, $\sqrt{72}$ and $2\sqrt{18}$ are not.
Like surdsSurds with the same number under the root sign, which can be added or subtracted. Like this: $3\sqrt{5}$ and $2\sqrt{5}$ are like surds, so they add to $5\sqrt{5}$.
Coefficient of a surdThe number multiplying the root. Like this: in $7\sqrt{3}$ the coefficient is 7 and the surd part is $\sqrt{3}$.
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What a surd is, and how to simplify one
Work through the core explanation before applying it.
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What a surd is, and how to simplify one
core concept
A root is a surd when its value is irrational. $\sqrt{9} = 3$ is rational, so it is not a surd. $\sqrt{10} = 3.1622...$ never terminates or repeats, so it is.
To simplify, split the number under the root into a perfect square times whatever is left, using the rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$. For $\sqrt{72}$, the largest perfect square factor is 36, giving $\sqrt{36}\sqrt{2} = 6\sqrt{2}$.
If you use a smaller square you still get there, just more slowly: $\sqrt{72} = \sqrt{4}\sqrt{18} = 2\sqrt{18} = 2 \times 3\sqrt{2} = 6\sqrt{2}$.
Roots do not split over addition. $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$. The rule works for multiplication and division only.
Quick check: simplify $\sqrt{50}$.
A surd is an irrational root. Simplify by writing the number under the root as a perfect square times a remainder, then taking the square root of the perfect square outside. Roots split over multiplication and division, never over addition.
Pause, copy the definition of a surd, the simplifying method with $\sqrt{72} = 6\sqrt{2}$, and the counter-example proving $\sqrt{a+b} \neq \sqrt{a}+\sqrt{b}$, into your book.
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Multiplying and dividing surds
core concept
We just saw how to simplify a single surd by extracting perfect squares. That raises a question: what happens when two surds are multiplied or divided? This card answers it → multiply the coefficients and the surd parts separately, then simplify the result.
Multiply the numbers outside the roots together, and the numbers inside the roots together: $(2\sqrt{3})(5\sqrt{6}) = 10\sqrt{18}$.
Then simplify what you get: $10\sqrt{18} = 10 \times 3\sqrt{2} = 30\sqrt{2}$. Almost every product needs this final simplification step.
Division works the same way: $\dfrac{12\sqrt{20}}{3\sqrt{5}} = 4\sqrt{4} = 8$. A surd divided by a surd can come out rational.
Squaring a surd. $\left(\sqrt{a}\right)^2 = a$, so $\left(3\sqrt{5}\right)^2 = 9 \times 5 = 45$. Square the coefficient and the surd separately.
Which expression does NOT equal $6\sqrt{2}$?
Multiply coefficients together and the numbers under the roots together, then simplify. Divide the same way. $\left(\sqrt{a}\right)^2 = a$, so squaring removes the root.
Pause, copy the multiply rule with the worked $(2\sqrt{3})(5\sqrt{6}) = 30\sqrt{2}$, the divide example $\frac{12\sqrt{20}}{3\sqrt{5}} = 8$, and the squaring rule, into your book.
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Adding, subtracting and expanding brackets
core concept
We just saw that surds multiply and divide freely. That raises a question: can you add them just as freely? This card answers it → no, only like surds combine, so you must simplify everything first to see which are alike.
Only like surds add or subtract. Treat $\sqrt{5}$ as if it were a variable: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$, but $3\sqrt{5} + 2\sqrt{7}$ stays as it is.
Simplify before deciding. $\sqrt{8} + \sqrt{18}$ looks unlike, but simplifies to $2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$.
Squaring a binomial surd. $(\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}$. The middle term is the one students lose, exactly as with $(a+b)^2$.
Fill the blank: $\sqrt{12} + \sqrt{27}$ simplifies to $5\sqrt{\;}$ where the number under the root is .
Only like surds add or subtract, so always simplify first to reveal which are alike. Brackets containing surds expand exactly as algebraic brackets, including the middle term when squaring a binomial.
Pause, copy the like-surds rule, the worked $\sqrt{8} + \sqrt{18} = 5\sqrt{2}$, and one bracket expansion such as $(\sqrt{3}+1)^2 = 4 + 2\sqrt{3}$, into your book.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · SIMPLIFYING
Simplify $\sqrt{72} + \sqrt{50} - \sqrt{18}$.
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$\sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$
Largest perfect square factor of 72 is 36.
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$\sqrt{50} = 5\sqrt{2}$ and $\sqrt{18} = 3\sqrt{2}$
Perfect square factors 25 and 9.
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$6\sqrt{2} + 5\sqrt{2} - 3\sqrt{2} = 8\sqrt{2}$
All three are like surds now, so combine the coefficients.
PROBLEM 2 · MULTIPLYING
Simplify $(2\sqrt{3})(5\sqrt{6})$.
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Coefficients: $2 \times 5 = 10$
Multiply the numbers outside the roots.
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Surds: $\sqrt{3} \times \sqrt{6} = \sqrt{18}$
Multiply the numbers under the roots.
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$10\sqrt{18} = 10 \times 3\sqrt{2} = 30\sqrt{2}$
Simplify $\sqrt{18}$, then multiply out.
PROBLEM 3 · EXPANDING BRACKETS
Expand and simplify $(\sqrt{5} + 2)(\sqrt{5} - 3)$.
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$= \sqrt{5}\sqrt{5} - 3\sqrt{5} + 2\sqrt{5} - 6$
Multiply every term by every term, as with any binomial product.
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$\sqrt{5}\sqrt{5} = 5$
A surd times itself gives the number under the root.
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$= 5 - \sqrt{5} - 6 = -1 - \sqrt{5}$
Collect the like surds and the rational terms separately.
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Quick-fire practice
Work through the core explanation before applying it.
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Quick-fire practice
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Simplify $\sqrt{48}$.
Simplify $3\sqrt{7} - \sqrt{7}$.
Simplify $(4\sqrt{2})(3\sqrt{5})$.
Expand $(\sqrt{2} + 3)^2$.
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Revisit your estimate of $\sqrt{50}$
Run the quick drill and copy the summary into your book.
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Revisit your estimate of $\sqrt{50}$
At the start you estimated $\sqrt{50}$. Now write it in simplified surd form and explain how the simplified form makes the estimate easy to check mentally.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
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Short answer
Write full responses, then check them against the model answers.
Q3 (2 marks): $\sqrt{16} = 4$, a whole number, so it is rational and not a surd; $\sqrt{17}$ is irrational, so it is a surd [1]. A surd is in simplified form when the number under the root has no perfect square factor other than 1 [1].
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Review and finish
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Boss battle · Surd Simplifier
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Simplify, combine and expand surd expressions at speed without reaching for a decimal. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.