Unit 3 "How-To" Guide

Unit 3 "How-To" Guide: Radical Expressions

Reference Sheet

Day 1: How to Simplify Radicals

  1. Find the largest perfect square (or cube) that divides into the number under the radical.
  2. Rewrite the radical as a product of two radicals.
  3. Take the square root (or cube root) of the perfect part. This number goes outside.
  4. Leave the other number under the radical.

Day 2: How to Add, Subtract & Multiply

Add/Subtract:

  1. Simplify all radicals first.
  2. Combine "like radicals" (same number inside) by adding/subtracting the coefficients (numbers outside).

Multiply:

  1. Multiply the coefficients.
  2. Multiply the numbers under the radical.
  3. Simplify the final radical.

Day 3: Radicals & Fractional Exponents

\( \sqrt[n]{x^m} = x^{m/n} \)

  1. The index of the root is the denominator of the fraction.
  2. The power on the number is the numerator.

Day 4: How to Rationalize the Denominator

  1. Identify the radical in the denominator.
  2. Multiply the top and bottom of the fraction by that same radical.
  3. Simplify. The denominator should no longer have a radical.

Worked-Out Examples

Day 1: Simplifying Radicals

Simplify: \( \sqrt{72} \)

\(\rightarrow \sqrt{36 \cdot 2}\)

\(\rightarrow \sqrt{36} \cdot \sqrt{2}\)

\(\rightarrow 6\sqrt{2}\)

Simplify: \( \sqrt[3]{54} \)

\(\rightarrow \sqrt[3]{27 \cdot 2}\)

\(\rightarrow \sqrt[3]{27} \cdot \sqrt[3]{2}\)

\(\rightarrow 3\sqrt[3]{2}\)

Day 2: Operations with Radicals

Subtract: \( \sqrt{50} - \sqrt{8} \)

\(\rightarrow \sqrt{25 \cdot 2} - \sqrt{4 \cdot 2}\)

\(\rightarrow 5\sqrt{2} - 2\sqrt{2}\)

\(\rightarrow 3\sqrt{2}\)

Multiply: \( 3\sqrt{2} \cdot 4\sqrt{6} \)

\(\rightarrow (3 \cdot 4)\sqrt{2 \cdot 6} = 12\sqrt{12}\)

\(\rightarrow 12\sqrt{4 \cdot 3} = 12 \cdot 2\sqrt{3}\)

\(\rightarrow 24\sqrt{3}\)

Day 3: Radicals & Fractional Exponents

Radical to Exponent:

Convert \( \sqrt[3]{x^2} \) to exponential form.

\(\rightarrow x^{2/3}\)

Exponent to Radical:

Convert \( y^{3/4} \) to radical form.

\(\rightarrow \sqrt[4]{y^3}\)

Day 4: Rationalizing the Denominator

Example 1: Basic

\( \frac{2}{\sqrt{5}} \)

\(\rightarrow \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}\)

\(\rightarrow \frac{2\sqrt{5}}{5}\)

Example 2: Simplify First

\( \frac{5}{\sqrt{12}} \)

\(\rightarrow \frac{5}{\sqrt{4 \cdot 3}} = \frac{5}{2\sqrt{3}}\)

\(\rightarrow \frac{5}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{2 \cdot 3}\)

\(\rightarrow \frac{5\sqrt{3}}{6}\)