Grade 11 · MCR3U · Functions

MCR3U: Simplifying Radicals

A radical is in simplest form when the number under the root has no perfect square factor and no root is left in a denominator. You need this in the quadratics unit because the quadratic formula hands you answers like $\frac{4 \pm \sqrt{48}}{2}$ , and tests want them exact and simplified: $2 \pm 2\sqrt{3}$ .

How It Works

Pull out perfect squares. Split the number under the root into its largest perfect square factor times the rest, then take the square root of the square: $\sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$ . If you miss the largest one, keep going: $\sqrt{72} = 3\sqrt{8} = 3 \cdot 2\sqrt{2}$ ends in the same place.

Add only like radicals. $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$ , the way $3x + 5x = 8x$ , but $\sqrt{2} + \sqrt{3}$ does not combine. Simplify every radical first, because radicals that look different are often alike underneath: $\sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2}$ .

Multiply coefficient by coefficient and root by root. $2\sqrt{3} \times 5\sqrt{6} = 10\sqrt{18} = 30\sqrt{2}$ . Expanding brackets works the same way it does with variables, using $\sqrt{a} \times \sqrt{a} = a$ .

Rationalize a denominator. Multiply the top and bottom by the root in the denominator: $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$ .

Use the conjugate for a binomial denominator. For $\frac{1}{3 + \sqrt{2}}$ , multiply the top and bottom by $3 - \sqrt{2}$ . The bottom becomes a difference of squares, $9 - 2 = 7$ , and the root is gone.

Simplify the quadratic formula's answer. Simplify the root, then divide every term of the numerator by the common factor. Do not divide only the first term: $\frac{4 \pm 4\sqrt{3}}{2}$ is $2 \pm 2\sqrt{3}$ , not $2 \pm 4\sqrt{3}$ .

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Simplify $\sqrt{75}$ and $-2\sqrt{80}$ .
Show solution
1. $\sqrt{75} = \sqrt{25}\sqrt{3} = 5\sqrt{3}$ .
2. $-2\sqrt{80} = -2\sqrt{16}\sqrt{5} = -2(4)\sqrt{5} = -8\sqrt{5}$ .
Answer: $5\sqrt{3}$ and $-8\sqrt{5}$ .
Easy
Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8}$ .
Show solution
1. Simplify each: $5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2}$ .
2. All three are like radicals, so collect the coefficients: $5 + 3 - 2 = 6$ .
Answer: $6\sqrt{2}$
Medium
Expand and simplify $(2\sqrt{3} - \sqrt{2})(\sqrt{3} + 4\sqrt{2})$ .
Show solution
1. Multiply each term by each term: $2\sqrt{3} \cdot \sqrt{3} + 2\sqrt{3} \cdot 4\sqrt{2} - \sqrt{2} \cdot \sqrt{3} - \sqrt{2} \cdot 4\sqrt{2}$ .
2. That is $6 + 8\sqrt{6} - \sqrt{6} - 8$ .
3. Collect: $-2 + 7\sqrt{6}$ .
Answer: $7\sqrt{6} - 2$
Medium
Solve $x^2 + 6x - 9 = 0$ , giving the roots in simplest radical form.
Show solution
1. Quadratic formula: $x = \frac{-6 \pm \sqrt{36 + 36}}{2} = \frac{-6 \pm \sqrt{72}}{2}$ .
2. $\sqrt{72} = 6\sqrt{2}$ , so $x = \frac{-6 \pm 6\sqrt{2}}{2}$ .
3. Divide both terms of the numerator by $2$ .
Answer: $x = -3 \pm 3\sqrt{2}$
Test level
Rationalize the denominator and simplify $\frac{4 + \sqrt{3}}{2 - \sqrt{3}}$ .
Show solution
1. Multiply the top and bottom by the conjugate, $2 + \sqrt{3}$ .
2. Top: $(4 + \sqrt{3})(2 + \sqrt{3}) = 8 + 4\sqrt{3} + 2\sqrt{3} + 3 = 11 + 6\sqrt{3}$ .
3. Bottom: $(2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1$ .
Answer: $11 + 6\sqrt{3}$

Learn It on Martian Lab

Martian Lab teaches this in the Radicals lesson of its MCR3U Rational Expressions unit. Watch it, then take the Quadratic Functions practice tests to see if it stuck.

Watch the Radicals lesson free

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Free Quadratic Functions practice test, with solutions

More Quadratic Functions topics: Max/Min and Vertex Form · Linear-Quadratic Systems · Quadratic Word Problems

More MCR3U topics: Inverse Functions · Transformations of Functions · Sine Law and the Ambiguous Case · Domain, Range and Function Notation · Simplifying Rational Expressions · Operations with Rational Expressions · Exponent Laws and Rational Exponents · Graphing Exponential Functions · Exponential Growth and Decay · Special Angles and the CAST Rule · Cosine Law and 3D Problems · Transformations of Sine and Cosine · Sinusoidal Word Problems · Arithmetic Sequences and Series · Geometric Sequences and Series · Pascal's Triangle and the Binomial Theorem · Simple and Compound Interest · Annuities: Future and Present Value

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