Find the following roots.

1. \(\sqrt{36}\)

2. \(-\sqrt{36}\)

3. \(\sqrt{\dfrac{1}{36}}\)

Find the following roots, if possible

1. \(\sqrt[3]{-8}\)

2. \(\sqrt[3]{-64}\)

3. \(\sqrt{-4}\)

4. \(-\sqrt{4}\)

5. \(\sqrt[4]{-16}\)

6. \(-\sqrt[4]{16}\)

Simplify each expression.

1. \(\sqrt{36}+\sqrt{64}\)

2. \(\sqrt{36+64}\)

3. \(\dfrac{3\sqrt{5}}{6}\)

4. \(\dfrac{4+6\sqrt{3}}{4}\)

A tent pole is \(8\) feet long and makes a \(90^{\circ}\) angle with the ground. One end of a rope is attached to the top of the pole, and the other end is anchored to the ground \(6\) feet from the bottom of the pole. Find the length of the rope.

Additional Problems
Simplify each expression

1. \(\displaystyle\frac{3\sqrt{5}}{6}\)

2. \(\displaystyle\frac{4\sqrt{2}}{2}\)

3. \(\displaystyle\frac{4+6\sqrt{3}}{4}\)

4. \(\displaystyle\frac{10+5\sqrt{6}}{5}\)

Mini Lecture
Simplify.

1. \(\sqrt{9}\)

2. \(-\sqrt{9}\)

3. \(\sqrt{-25}\)

4. \(-\sqrt{64}\)

5. \(\sqrt[3]{-8}\)

6. \(\sqrt{9x^2}\)

7. \(\sqrt{36a^6}\)

Enrichment
Constructing a Golden Rectangle from a square of side length \(2\).

Simplify \(\sqrt{\displaystyle\frac{50}{49}}\)

Simplify \(\sqrt{\displaystyle\frac{12x^2}{25}}\), assume \(x\geq 0\).

Simplify each expression

1. \(\displaystyle\frac{\sqrt{12}}{6}\)

2. \(\displaystyle\frac{5\sqrt{18}}{15}\)

3. \(\displaystyle\frac{6+\sqrt{8}}{2}\)

4. \(\displaystyle\frac{4+\sqrt{36}}{2}\)

5. \(\displaystyle\frac{-2+\sqrt{48}}{4}\)

Simplify each expression

1. \(\displaystyle\frac{-4-\sqrt{40}}{4}\)

2. \(\displaystyle\frac{-6+\sqrt{48}}{2}\)

3. \(\displaystyle\frac{6-\sqrt{108}}{18}\)

Construct a golden rectangle from a square of side length \(4\). Then show that the ratio of the length to the width is the golden ratio \(\displaystyle\frac{1+\sqrt{5}}{2}\).

Additional Problems
Simplify each expression

1. \(\displaystyle\frac{\sqrt{12}}{6}\)

2. \(\displaystyle\frac{5\sqrt{18}}{15}\)

3. \(\displaystyle\frac{6+\sqrt{8}}{2}\)

4. \(\displaystyle\frac{-1+\sqrt{45}}{2}\)

5. \(\displaystyle\frac{4+\sqrt{36}}{2}\)

Mini Lecture
Simplify.

1. \(\sqrt{12}\)

2. \(\sqrt[3]{24}\)

3. \(5\sqrt{80}\)

4. \(3\sqrt{50xy^2}\)

5. \(\sqrt{\displaystyle\frac{128}{49}}\)

Roots and Radicals 5: Constructing a Golden Rectangle from a Square of Side 4

No matter how large or small a golden rectangle is, the ratio of length to width is always the same.

Combine \(3\sqrt{5}-7\sqrt{5}\)

Combine \(7\sqrt{2}-3\sqrt{2}+6\sqrt{2}\)

Simplify: \(6\sqrt{5}+5\sqrt{2}-3\sqrt{5}\).

Combine \(3\sqrt{50}+2\sqrt{32}\)

Combine terms in the expression \(\sqrt[3]{375}+\sqrt[3]{81}-6\sqrt[3]{3}\).

Simplify \(a\sqrt{12}+5\sqrt{3a^2}\), assume \(a\geq0\).

Combine \(\sqrt{20x^3}-3x\sqrt{45x}+10\sqrt{25x^2}\), assume \(x\geq0\).

Simplify \(\displaystyle\frac{6+\sqrt{12}}{4}\)

Mini Lecture
Simplify.

1. \(3\sqrt{2}+4\sqrt{2}\)

2. \(\sqrt{8}+2\sqrt{2}\)

3. \(\displaystyle\frac{1}{5}\sqrt{75}-\displaystyle\frac{1}{2}\sqrt{12}\)

4. \(6\sqrt{48}-2\sqrt{12}+5\sqrt{27}\)

Multiply: \(\sqrt{2}\cdot\sqrt{14}\)

Multiply \(\left(3\sqrt{5}\right)\left(2\sqrt{7}\right)\)

Multiply \(\sqrt{5}\left(\sqrt{2}+\sqrt{5}\right)\)

Use the power property for radicals to simplify the following radical expressions: \(\left(\sqrt{11}\right)^2\)

Use the power property for radicals to simplify the following radical expression: \(\left(\sqrt[3]{-6}\right)^3\)

Use the power property for radicals to simplify the following radical expression: \(\left(\sqrt[4]{2}\right)^4\)

Use the power property for radicals to simplify the following radical expression: \(\left(\sqrt[5]{3}\right)^5\)

Multiply: \(3\sqrt{2}\left(2\sqrt{5}+5\sqrt{6}\right)\)

Multiply \(\left(\sqrt{5}+2\right)\left(\sqrt{5}+7\right)\)

Multiply \(\left(\sqrt{x}+3\right)\left(\sqrt{x}-7\right)\)

Simplify \(\left(\sqrt{3}-2\right)^2\)

Multiply: \(\left(\sqrt{6}+\sqrt{3}\right)^2\)

Multiply \(\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)\)

Multiply \(\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)\)

Multiply: \(\sqrt[3]{9}\cdot\sqrt[3]{3}\)

Multiply: \(\left(\sqrt[3]{4}-5\right)^2\)

Divide \(\dfrac{\sqrt{60}}{\sqrt{3}}\)

Divide: \(\dfrac{6\sqrt{20}}{2\sqrt{5}}\)

Divide: \(\dfrac{5\sqrt[3]{48}}{4\sqrt[3]{2}}\)

Simplify \(\sqrt{\displaystyle\frac{1}{2}}\)

Simplify \(\sqrt{\displaystyle\frac{2}{3}}\)

Simplify: \(\sqrt{\dfrac{4x^3y^2}{3z}}\)

Rationalize the denominator \(\dfrac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}\)

Rationalize the denominator \(\dfrac{2}{5-\sqrt{3}}\)

Rationalize the denominator in the expression \(\dfrac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)

Simplify each expression.

1. \(\sqrt[3]{\dfrac{2}{3}}\)

2. \(\sqrt[3]{\dfrac{3x^4}{4y^2z}}\)

Mini Lecture
Simplify.

1. \(\left(2\sqrt{3}\right)\left(5\sqrt{7}\right)\)

2. \(\sqrt{3}\left(2\sqrt{2}+\sqrt{3}\right)\)

3. \(\left(\sqrt{x}+3\right)^2\)

4. \(\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)\)

5. Divide \(\dfrac{\sqrt{3}}{\sqrt{5}-\sqrt{2}}\)

Enrichment
Every Golden Rectangle contains an infinite number of other Golden Rectangles. The key to seeing this is rationalizing the denominator.

Roots and Radicals 8: The Golden Ratio in Bagdad in 1150.

Here is one problem that has almost everything we know about roots and radicals contained within it.

Solve \(\sqrt{x+1}=7\)

Solve \(\sqrt{2x-3}=-9\)

Solve \(\sqrt{3a-2}+3=5\)

Solve \(\sqrt{x+15}=x+3\)

Solve \(\sqrt{2x+1}=\sqrt{x-4}+3\)

Solve: \(\sqrt[3]{3-4x}+2=5\)

Enrichment
Understanding the squaring property of equality.

Mini Lecture
Solve.

1. \(\sqrt{x+1}=2\)

2. \(\sqrt{x-9}=-6\)

3. \(\sqrt{3x+4}-3=2\)

4. \(\sqrt{x+3}=x-3\)