We simplify radicals using the product/quotient rule for radicals. This rule allows us to break the radicand up and pull out rational numbers. For example, we can break up the square root of 20 into: the square root of 5 times the square root of 4. The square root of 4 represents a rational number 2. We report our simplified radical as 2 times the square root of 5.

Test Objectives
• Demonstrate the ability to use the product rule for radicals
• Demonstrate the ability to use the quotient rule for radicals
• Demonstrate the ability to simplify a radical

#1:

Instructions: Simplify each, all variables are positive real numbers.

a) $$\sqrt[3]{-625}$$

b) $$\sqrt[5]{96}$$

c) $$\sqrt[4]{162}$$

#2:

Instructions: Simplify each, all variables are positive real numbers.

a) $$-2\sqrt{147p^4}$$

b) $$-5\sqrt{90x^4}$$

#3:

Instructions: Simplify each, all variables are positive real numbers.

a) $$3\sqrt[5]{192x^7y^4z^8}$$

b) $$6\sqrt[4]{160x^4yz^9}$$

#4:

Instructions: Simplify each, all variables are positive real numbers.

a) $$\frac{4\sqrt[4]{20}}{3\sqrt[4]{1024}}$$

b) $$\frac{3\sqrt[5]{-2}}{5\sqrt[5]{3125}}$$

#5:

Instructions: Simplify each, all variables are positive real numbers.

a) $$\sqrt{10} \cdot \sqrt[6]{9}$$

Written Solutions:

#1:

Solutions:

a) $$-5\sqrt[3]{5}$$

b) $$2\sqrt[5]{3}$$

c) $$3\sqrt[4]{2}$$

#2:

Solutions:

a) $$-14p^2\sqrt{3}$$

b) $$-15x^2\sqrt{10}$$

#3:

Solutions:

a) $$6xz\sqrt[5]{6x^2y^4z^3}$$

b) $$12xz^2\sqrt[4]{10yz}$$

#4:

Solutions:

a) $$\frac{\sqrt[4]{5}}{3}$$

b) $$\frac{-3\sqrt[5]{2}}{25}$$

#5:

Solutions:

a) $$\sqrt[6]{9000}$$