Operations with radicals

We are going to simplify expressions by grouping like radicals. Imagine that $\sqrt{3}$ is like an "x" or an "apple". Carry out the following additions and subtractions:

1. $5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2}$

2. $7\sqrt[3]{5} - 2\sqrt[3]{5}$

3. $4\sqrt{6} + \sqrt{6} - 8\sqrt{6}$

To add and subtract radicals, they must be like radicals (same index and same radicand). Simplify the following radicals by expressing the numbers as products of prime factors, then calculate:

1. a) $3\sqrt{8} - 5\sqrt{2} + \sqrt{50}$
2. b) $\sqrt{12} + \sqrt{75} - \sqrt{27}$
3. c) $2\sqrt[3]{16} - \sqrt[3]{54}$

Use the property of the product of roots with the same index ($\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$) to group these operations into a single root and, if possible, work out the exact final result:

• $\sqrt{2} \cdot \sqrt{32}$

• $\sqrt[3]{2} \cdot \sqrt[3]{4}$

• $\sqrt{5} \cdot \sqrt{20}$

• $\sqrt{3} \cdot \sqrt{27}$

Carry out the following multiplications and divisions of radicals. Remember that they must have the same index to operate. If possible, simplify the final result by extracting factors from the root.

1. a) $\sqrt{18} \cdot \sqrt{2}$
2. b) $\sqrt[3]{4} \cdot \sqrt[3]{16}$
3. c) $\frac{\sqrt{75}}{\sqrt{3}}$
4. d) $\sqrt{2x} \cdot \sqrt{8x^3}$

Calculate the perimeter and area of a rectangular plot of land whose dimensions are given by radicals. The base measures $\sqrt{27}$ metres and the height $\sqrt{3}$ metres. Simplify the results as much as possible.

When we have a root inside another root, we can simplify it by multiplying their indices ($\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a}$). Simplify the following expressions to a single root:

• $\sqrt{\sqrt{81}}$

• $\sqrt[3]{\sqrt{64}}$

• $\sqrt{\sqrt[3]{x}}$

Fractional powers are another way of writing radicals. Use this relationship to simplify the following expressions and give the final answer in radical form:

1. a) Write $5^{3/4}$ in radical form.
2. b) Write $\sqrt[5]{2^3}$ in index form.
3. c) Calculate by first expressing as a power: $\sqrt[3]{\sqrt{64}}$
4. d) Simplify: $\frac{\sqrt[4]{x^3}}{\sqrt{x}}$

Apply the rules of radicals (root of a root and power of a root) to simplify the following expressions to a single irreducible radical:

1. a) $\sqrt{\sqrt[3]{64}}$
2. b) $(\sqrt[3]{2})^6$
3. c) $\sqrt{2 \cdot \sqrt{2}}$ (Hint: Move the first 2 inside the second square root)

You are going to buy a new television, and the dimensions are given in inches along the diagonal. The screen is rectangular, with a width of 40 cm and a height of 30 cm. Using Pythagoras' theorem ($h = \sqrt{a^2 + b^2}$), calculate the length of the screen's diagonal in centimetres.

Note: The result is an exact root.

We want to attach a guy wire to support a vertical post 4 metres high. The wire will be anchored to the ground at a distance of $\sqrt{20}$ metres from the base of the post. Calculate the exact length of cable needed using Pythagoras' theorem. Express the answer as a simplified radical.

We have a cubic water tank. We know that its total volume is $128 \text{ m}^3$. Calculate the length of the edge of the tank. Express the answer as a simplified radical (by taking factors out of the root) and do not use decimals.

(Hint: Remember that the volume of a cube is $V = \text{arista}^3$)