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.

Assume all variables are positive real numbers.

$$a)\hspace{.2em}\sqrt{96x}$$

$$b)\hspace{.2em}\sqrt{180x^3}$$

#2:

Instructions: Simplify each.

Assume all variables are positive real numbers.

$$a)\hspace{.2em}\sqrt[7]{512x^2}$$

$$b)\hspace{.2em}\sqrt[4]{80x^7}$$

#3:

Instructions: Simplify each.

Assume all variables are positive real numbers.

$$a)\hspace{.2em}4\sqrt[3]{320x^8y^3}$$

$$b)\hspace{.2em}-6\sqrt[3]{64xy^5}$$

#4:

Instructions: Simplify each.

Assume all variables are positive real numbers.

$$a)\hspace{.2em}-2\sqrt[3]{375x^2y^4z^8}$$

$$b)\hspace{.2em}-5\sqrt{72x^4yz^2}$$

#5:

Instructions: Simplify each.

Assume all variables are positive real numbers.

$$a)\hspace{.2em}\sqrt[5]{\sqrt[9]{2x}}$$

$$b)\hspace{.2em}\sqrt[5]{\sqrt{x + 1}}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}4 \sqrt{6x}$$

$$b)\hspace{.2em}6x \sqrt{5x}$$

#2:

Solutions:

$$a)\hspace{.2em}2 \sqrt[7]{4x^2}$$

$$b)\hspace{.2em}2x \sqrt[4]{5x^3}$$

#3:

Solutions:

$$a)\hspace{.2em}16x^2y\sqrt[3]{5x^2}$$

$$b)\hspace{.2em}-24y\sqrt[3]{xy^2}$$

#4:

Solutions:

$$a)\hspace{.2em}-10yz^2\sqrt[3]{3x^2yz^2}$$

$$b)\hspace{.2em}-30x^2z\sqrt{2y}$$

#5:

Solutions:

$$a)\hspace{.2em}\sqrt[45]{2x}$$

$$b)\hspace{.2em}\sqrt[10]{x + 1}$$