Unit 3 Resources & Lessons
Use the buttons below to access unit-wide resources or jump to a specific lesson. (SOL: A.EO.4a-d)
Daily Lesson Slides & Worksheets
Day 1: Simplify Square & Cube Roots
Day 2: Rational Exponents
Day 3: Add/Subtract/Multiply Radicals
Day 4: Equivalency of Radicals
Unit 3: Study Guide
Day 1: Simplify Square & Cube Roots
Objective: To simplify radical expressions, including square roots and cube roots of numbers and variables.
Concept
To simplify a radical, we find the largest perfect square (or perfect cube) that divides the number under the radical (radicand). We then split the radical into two and take the root of the perfect part.
I Do: Examples
Example 1 (Square Root): Simplify \(\sqrt{72}\).
Step 1: Find the largest perfect square that divides 72. (Perfect squares: 4, 9, 16, 25, 36, 49...)
\(72 = 36 \cdot 2\)
Step 2: Split the radical and simplify.
\(\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2}\)
\(6\sqrt{2}\)
Example 2 (Cube Root): Simplify \(\sqrt[3]{54}\).
Step 1: Find the largest perfect cube that divides 54. (Perfect cubes: 8, 27, 64...)
\(54 = 27 \cdot 2\)
Step 2: Split the radical and simplify.
\(\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = \sqrt[3]{27} \cdot \sqrt[3]{2}\)
\(3\sqrt[3]{2}\)
We Do: Let's Try Together
You Do: Practice On Your Own
Practice
We Do 1: \(\sqrt{50}\) =
We Do 2: \(\sqrt[3]{16}\) =
You Do 1: Simplify: \(\sqrt{98}\) =
You Do 2: Simplify: \(\sqrt[3]{250}\) =
Day 2: Rational Exponents
Objective: To convert expressions between radical form and rational exponent form, and to evaluate them.
Concept
A radical can be rewritten as an exponent that is a fraction. The key is "Power over Root". The exponent of the radicand is the numerator, and the index of the root is the denominator.
I Do: Examples
Example 1 (Convert to rational exponent): Write \(\sqrt[5]{x^3}\) in rational exponent form.
The power is 3, the root is 5.
\(x^{3/5}\)
Example 2 (Convert to radical): Write \(y^{2/3}\) in radical form.
The power is 2, the root is 3.
\(\sqrt[3]{y^2}\)
Example 3 (Evaluate): Evaluate \(8^{2/3}\).
Step 1: Rewrite as a radical: \(\sqrt[3]{8^2}\) or \((\sqrt[3]{8})^2\). It's easier to take the root first.
\((\sqrt[3]{8})^2 = (2)^2\)
\(4\)
We Do: Let's Try Together
You Do: Practice On Your Own
Practice
We Do 1: Rewrite \(\sqrt{b^5}\) with a rational exponent. =
We Do 2: Evaluate \(16^{3/4}\). =
You Do 1: Write \(c^{7/4}\) in radical form. =
You Do 2: Evaluate \(25^{3/2}\). =
Day 3: Add/Subtract/Multiply Radicals
Objective: To determine the sums, differences, and products of radical expressions.
Concept
Adding & Subtracting
You can only add or subtract radicals that are like terms. This means they must have the same index and the same radicand. Sometimes, you must simplify first!
Multiplying
To multiply radicals, multiply the coefficients (outsides) and multiply the radicands (insides). Then, simplify the result.
\((a\sqrt{b}) \cdot (c\sqrt{d}) = ac\sqrt{bd}\)
I Do: Examples
Example 1 (Add with simplifying): Simplify \(\sqrt{12} + \sqrt{27}\).
Step 1: Simplify both radicals. \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\). And \(\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}\).
Step 2: Now they are like terms. Combine them. \(2\sqrt{3} + 3\sqrt{3}\)
\(5\sqrt{3}\)
Example 2 (Multiply with simplifying): Simplify \(\sqrt{8} \cdot \sqrt{6}\).
Step 1: Multiply the radicands. \(\sqrt{8 \cdot 6} = \sqrt{48}\).
Step 2: Simplify the result. The largest perfect square in 48 is 16. \(\sqrt{48} = \sqrt{16 \cdot 3}\)
\(4\sqrt{3}\)
We Do: Let's Try Together
You Do: Practice on Your Own
Practice
We Do 1: \(8\sqrt{5} - 3\sqrt{5}\) =
We Do 2: \(2\sqrt{3} \cdot 4\sqrt{5}\) =
You Do 1: Simplify: \(\sqrt{18} + \sqrt{32}\) =
You Do 2: Simplify: \(5\sqrt{2} \cdot 3\sqrt{10}\) =
Day 4: Equivalency of Radicals
Objective: To write equivalent radical expressions by rationalizing the denominator.
Concept: Rationalizing the Denominator
A radical expression is not considered fully simplified if there is a radical in the denominator. To fix this, we rationalize the denominator. We multiply both the numerator and the denominator by the radical from the denominator. This is like multiplying by 1, so it doesn't change the value.
I Do: Examples
Example 1: Rationalize the denominator of \(\frac{5}{\sqrt{3}}\).
Step 1: Multiply the numerator and denominator by \(\sqrt{3}\).
\(\frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{\sqrt{9}}\)
Step 2: Simplify the denominator.
\(\frac{5\sqrt{3}}{3}\)
Example 2: Simplify \(\frac{6}{\sqrt{18}}\).
Step 1: Simplify the radical in the denominator first. \(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\).
\(\frac{6}{3\sqrt{2}}\)
Step 2: Reduce the fraction outside the radical. \(\frac{6}{3} = 2\). So we have \(\frac{2}{\sqrt{2}}\).
Step 3: Rationalize the denominator. \(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2}\)
\(\sqrt{2}\)
We Do: Let's Try Together
You Do: Practice on Your Own
Practice
We Do 1: Rationalize: \(\frac{1}{\sqrt{2}}\) =
We Do 2: Rationalize: \(\frac{10}{\sqrt{5}}\) =
You Do 1: Simplify completely: \(\sqrt{\frac{9}{10}}\) =
You Do 2: Simplify completely: \(\frac{15}{\sqrt{75}}\) =
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