Lesson Objectives
• Demonstrate an understanding of how to simplify a radical
• Learn how to multiply radical expressions
• Learn how to rationalize a binomial denominator
• Learn how to write radical expressions with quotients in lowest terms

## How to Perform Operations with Radicals

Now that we have a good understanding of radicals, we can look at some more challenging scenarios. Let's begin by restating the rules for the simplified form of a radical.

### Simplified Form of a Radical

• The radicand contains no factor (except 1) that is a:
• Perfect Square » Square Root
• Perfect Cube » Cube Root
• Perfect Fourth » Fourth Root
• So on and so forth...
• The radicand cannot contain fractions
• There is no radical present in any denominator

Example 1: Simplify each $$2\sqrt{10}\left(5 + 4\sqrt{5}\right)$$ To perform our multiplication, we will use the distributive property: $$2\sqrt{10}\left(5 + 4\sqrt{5}\right)=$$ $$2\sqrt{10} \cdot 5 + 2\sqrt{10} \cdot 4\sqrt{5} =$$ $$(2 \cdot 5)\sqrt{10} + (2 \cdot 4)\sqrt{10 \cdot 5} =$$ $$10\sqrt{10} + 8 \cdot \sqrt{25} \cdot \sqrt{2} =$$ $$10\sqrt{10} + (8 \cdot 5)\sqrt{2} =$$ $$10\sqrt{10} + 40\sqrt{2}$$ Example 2: Simplify each $$\left(-\sqrt{2} - 6\right)\left(3\sqrt{2} + 6\right)$$ Since we have two terms multiplied by two terms, we can use FOIL
First Terms: $$-\sqrt{2} \cdot 3\sqrt{2} =$$ $$(-1 \cdot 3) \sqrt{2 \cdot 2} =$$ $$-3 \cdot 2 = -6$$ Outer Terms: $$-\sqrt{2} \cdot 6 =$$ $$(-1 \cdot 6)\sqrt{2} =$$ $$-6\sqrt{2}$$ Inner Terms: $$-6 \cdot 3\sqrt{2} =$$ $$(-6 \cdot 3)\sqrt{2} =$$ $$-18\sqrt{2}$$ Last Terms: $$-6 \cdot 6 = -36$$ Combine Like Terms: $$-6 - 6\sqrt{2} - 18\sqrt{2} - 36 =$$ $$(-6 - 36) + (-6\sqrt{2} - 18\sqrt{2})=$$ $$-42 + (-6 - 18)\sqrt{2} =$$ $$-42 - 24\sqrt{2}$$
In our special products lesson, we learned about conjugates. Conjugates occur when we have the sum and difference of the same two terms. If we multiply conjugates together using FOIL, the O and I step cancel: $$(5x + 1)(5x - 1)$$ First Terms: $$5x \cdot 5x = 25x^2$$ Outer Terms: $$5x \cdot -1 = -5x$$ Inner Terms: $$1 \cdot 5x = 5x$$ Last Terms: $$1 \cdot -1 = -1$$ If we combine like terms, the O and I steps will cancel: $$\require{cancel}25x^2 + (-5x) + 5x - 1 =$$ $$\require{cancel}25x^2 + \cancel{(-5x)} + \cancel{5x} - 1 =$$ $$25x^2 - 1$$ When we simplify a radical expression with two terms in the denominator and at least one is a radical, we multiply the numerator and denominator by the conjugate of the denominator. Let's look at an example.
Example 3: Simplify each $$\frac{8}{6 + 2\sqrt{7}}$$ To rationalize the denominator, we will multiply the numerator and denominator by the conjugate of the denominator: $$\frac{8}{6 + 2\sqrt{7}} \cdot \frac{6 - 2\sqrt{7}}{6 - 2\sqrt{7}}$$ Our numerator: $$8(6 - 2\sqrt{7}) =$$ $$48 - 16\sqrt{7}$$ Our denominator: $$(6 + 2\sqrt{7})(6 - 2\sqrt{7})$$ If we use FOIL, the O and I steps will cancel. We just need to perform the F and L steps:
First Terms: $$6 \cdot 6 = 36$$ Last Terms: $$2\sqrt{7} \cdot -2\sqrt{7} =$$ $$(2 \cdot -2) \cdot (\sqrt{7 \cdot 7})$$ $$-4 \cdot 7 = -28$$ Combine Like Terms: $$36 - 28 = 8$$ Our Fraction becomes: $$\frac{48 - 16\sqrt{7}}{8}$$ We can simplify this fraction by factoring an 8 out of the numerator. This will cancel with the 8 in the denominator: $$\frac{8(6 - 2\sqrt{7})}{8} =$$ $$\frac{\cancel{8}(6 - 2\sqrt{7})}{\cancel{8}} =$$ $$6 - 2\sqrt{7}$$ Example 4: Simplify each $$\frac{4y}{9x^3 - 6y\sqrt{7x}}$$ To rationalize the denominator, we will multiply the numerator and denominator by the conjugate of the denominator: $$\frac{4y}{9x^3 - 6y\sqrt{7x}} \cdot \frac{9x^3 + 6y\sqrt{7x}}{9x^3 + 6y\sqrt{7x}}$$ Our numerator: $$4y(9x^3 + 6y\sqrt{7x}) =$$ $$4y \cdot 9x^3 + 4y \cdot 6y\sqrt{7x} =$$ $$36x^3y + 24y^2\sqrt{7x}$$ Our denominator: $$(9x^3 - 6y\sqrt{7x})(9x^3 + 6y\sqrt{7x})$$ If we use FOIL, the O and I steps will cancel. We just need to perform the F and L steps:
First Terms: $$9x^3 \cdot 9x^3 = 81x^6$$ Last Terms: $$-6y\sqrt{7x} \cdot 6y\sqrt{7x} =$$ $$(-6y \cdot 6y) \cdot \sqrt{7x \cdot 7x} =$$ $$-36y^2 \cdot 7x = -252xy^2$$ Combine Like Terms: $$81x^6 - 252xy^2$$ Our Fraction becomes: $$\frac{36x^3y + 24y^2\sqrt{7x}}{81x^6 - 252xy^2}$$ We can simplify this fraction by factoring a 3 out of the numerator and the denominator. We can then cancel this 3 between the numerator and denominator: $$\frac{3(12x^3y + 8y^2\sqrt{7x})}{3(27x^6 - 84xy^2)} =$$ $$\frac{\cancel{3}(12x^3y + 8y^2\sqrt{7x})}{\cancel{3}(27x^6 - 84xy^2)} =$$ $$\frac{12x^3y + 8y^2\sqrt{7x}}{27x^6 - 84xy^2}$$