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Operations with Radicals

20 min Radicals & Rational Exponents

Operations with Radicals

Once you can simplify radicals, you can add, subtract, multiply, and divide them. Adding and subtracting require like radicals — the same index and the same radicand — just as adding fractions requires like denominators.

Multiplying radicals is more flexible: you multiply the radicands together and simplify the result. Dividing radicals introduces rationalization — the technique of eliminating radicals from the denominator by multiplying by a clever form of 1.

This lesson covers all four operations and the conjugate technique for rationalizing denominators that contain binomial radical expressions.

Adding and Subtracting Radicals

Radicals can be added or subtracted only when they are like radicals — same index and same radicand.

Rule

$$a\sqrt{n} + b\sqrt{n} = (a + b)\sqrt{n}$$

This works just like combining like terms in algebra.

Example 1

Simplify $3\sqrt{5} + 7\sqrt{5}$.

$$3\sqrt{5} + 7\sqrt{5} = (3 + 7)\sqrt{5} = 10\sqrt{5}$$

Example 2

Simplify $\sqrt{12} + \sqrt{27}$.

Simplify each radical first:

$\sqrt{12} = 2\sqrt{3}$ and $\sqrt{27} = 3\sqrt{3}$.

$$2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$$

Example 3

Simplify $5\sqrt{18} - 2\sqrt{50}$.

$5\sqrt{18} = 5 \cdot 3\sqrt{2} = 15\sqrt{2}$ and $2\sqrt{50} = 2 \cdot 5\sqrt{2} = 10\sqrt{2}$.

$$15\sqrt{2} - 10\sqrt{2} = 5\sqrt{2}$$

Multiplying Radicals

Product Rule

$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$

Example 4

Multiply $\sqrt{6} \cdot \sqrt{15}$.

$$\sqrt{6} \cdot \sqrt{15} = \sqrt{90} = \sqrt{9 \cdot 10} = 3\sqrt{10}$$

Example 5

Expand $(3 + \sqrt{2})(5 - \sqrt{2})$.

Use FOIL:

$$15 - 3\sqrt{2} + 5\sqrt{2} - (\sqrt{2})^2 = 15 + 2\sqrt{2} - 2 = 13 + 2\sqrt{2}$$

Example 6

Expand $(\sqrt{5} + \sqrt{3})^2$.

$$(\sqrt{5})^2 + 2\sqrt{5}\sqrt{3} + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}$$

Conjugates and Difference of Squares

Definition

The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$. Their product eliminates the radical:

$$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$$

Example 7

Multiply $(4 + \sqrt{3})(4 - \sqrt{3})$.

$$4^2 - (\sqrt{3})^2 = 16 - 3 = 13$$

Dividing Radicals — Rationalizing the Denominator

Quotient Rule

$$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \qquad (b > 0)$$

Rationalizing

To remove a radical from the denominator, multiply numerator and denominator by:

The radical itself if the denominator is a single radical.
The conjugate if the denominator is a binomial with a radical.

Example 8

Rationalize $\dfrac{5}{\sqrt{3}}$.

$$\frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$$

Example 9

Rationalize $\dfrac{3}{2 + \sqrt{5}}$.

Multiply by the conjugate $2 - \sqrt{5}$:

$$\frac{3}{2 + \sqrt{5}} \cdot \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{3(2 - \sqrt{5})}{4 - 5} = \frac{6 - 3\sqrt{5}}{-1} = -6 + 3\sqrt{5}$$

Practice Problems

1. $4\sqrt{3} + 6\sqrt{3}$

2. $\sqrt{8} + \sqrt{32}$

3. $7\sqrt{5} - 2\sqrt{45}$

4. $\sqrt{3} \cdot \sqrt{21}$

5. $(2\sqrt{6})^2$

6. $(1 + \sqrt{7})(1 - \sqrt{7})$

7. $\dfrac{10}{\sqrt{5}}$

8. $\dfrac{4}{3 - \sqrt{2}}$

9. $(\sqrt{3} + 2)^2$

10. $3\sqrt{20} + 2\sqrt{45} - \sqrt{80}$

11. $\sqrt{7} \cdot \sqrt{14}$

12. $\dfrac{6}{\sqrt{2}}$

13. $(5 - \sqrt{3})(5 + \sqrt{3})$

14. $\dfrac{1}{\sqrt{5} + \sqrt{2}}$

15. $2\sqrt{75} - 3\sqrt{48} + \sqrt{27}$

Show Answer Key

1. $10\sqrt{3}$

2. $2\sqrt{2} + 4\sqrt{2} = 6\sqrt{2}$

3. $7\sqrt{5} - 6\sqrt{5} = \sqrt{5}$

4. $\sqrt{63} = 3\sqrt{7}$

5. $4 \cdot 6 = 24$

6. $1 - 7 = -6$

7. $2\sqrt{5}$

8. $\dfrac{4(3+\sqrt{2})}{7} = \dfrac{12+4\sqrt{2}}{7}$

9. $3 + 4\sqrt{3} + 4 = 7 + 4\sqrt{3}$

10. $6\sqrt{5} + 6\sqrt{5} - 4\sqrt{5} = 8\sqrt{5}$

11. $\sqrt{98} = 7\sqrt{2}$

12. $3\sqrt{2}$

13. $25 - 3 = 22$

14. $\dfrac{\sqrt{5} - \sqrt{2}}{3}$

15. $10\sqrt{3} - 12\sqrt{3} + 3\sqrt{3} = \sqrt{3}$

Simplifying Radicals Rational Exponents

Operations with Radicals