Exercises: Operations with radicals

Work out the value of the following exact roots mentally. Remember that the index of the root tells you how many times you must multiply a number by itself to obtain the radicand.

• $\sqrt{81}$

• $\sqrt[3]{-27}$

• $\sqrt[4]{16}$

• $\sqrt{100}$

• $\sqrt[3]{125}$

Powers and roots are two different ways of writing the same thing. Complete the following table in your notebook by transforming the powers into roots and the roots into powers with a fractional exponent.

• Express as a root: $5^{1/2}$

• Express as a root: $3^{2/5}$

• Express as a power: $\sqrt[3]{7}$

• Express as a power: $\sqrt[4]{2^3}$

Let's simplify expressions by grouping like radicals. Imagine that $\sqrt{3}$ is like an "x" or an "apple". Perform 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. $4\sqrt{6} + \sqrt{6} - 8\sqrt{6}$

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, calculate the exact final result:

• $\sqrt{2} \cdot \sqrt{32}$

• $\sqrt[3]{2} \cdot \sqrt[3]{4}$

• $\sqrt{5} \cdot \sqrt{20}$

• $\sqrt{3} \cdot \sqrt{27}$

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:

• $\sqrt{\sqrt{81}}$

• $\sqrt[3]{\sqrt{64}}$

• $\sqrt{\sqrt[3]{x}}$

Sometimes the numbers inside the root have "leftover" factors that can be taken out. Extract all the factors you can from the following radicals.

Hint: First write the number as powers (example: $\sqrt{2^3} = \sqrt{2^2 \cdot 2} = 2\sqrt{2}$).

• $\sqrt{2^5}$

• $\sqrt{3^3}$

• $\sqrt{5^4}$

• $\sqrt[3]{2^4}$

We want to fence a square plot of land whose total area is $225 \text{ ft}^2$. To find out how many feet of fencing we need, we must first calculate the length of the side of the square.

1. Calculate the side length ($l$) knowing that the area is $l^2 = 225$.

2. Calculate the total perimeter of the plot.

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

Note: The result is an exact root.

Perform the following multiplications and divisions of radicals. Remember that to operate they must have the same index. 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) $\sqrt{2x} \cdot \sqrt{8x^3}$

Extract all the factors you can from the following radicals. Take into account the index of the root and the exponents of the radicands.

1. a) $\sqrt{2 \cdot 3^2 \cdot 5^5}$
2. b) $\sqrt[3]{81}$
3. c) $\sqrt{12x^3 y^4}$
4. d) $\sqrt[4]{32}$

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

To perform additions and subtractions with radicals, they must be similar (same index and same radicand). Simplify the following radicals by decomposing the numbers into prime factors and operate:

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

We have a cube-shaped water tank. We know that its total volume is $128 \text{ m}^3$. Calculate the length of the edge of the tank. Express the result as a simplified radical (extracting factors) and do not use decimals.

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

Powers with a fractional exponent are another way of writing radicals. Use this relationship to simplify the following expressions and give the final result in root form:

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

Apply the properties of the root of a root and the 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: Introduce the first 2 inside the second root)

We want to place a tension cable to support a vertical pole 4 feet high. The cable will be anchored to the ground at a distance of $\sqrt{20}$ feet from the base of the pole. Calculate the exact length of the cable needed using the Pythagorean theorem. Express the result as a simplified radical.

An architecture studio has designed a triangular decorative piece for a facade. The piece is an isosceles triangle (two equal sides and one unequal).

We know that the two equal sides measure $\sqrt{45}$ feet each, and the base (the unequal side) measures $\sqrt{72}$ feet.

Using the Pythagorean theorem and the properties of radicals (extracting factors and simplifying), calculate the exact height ($h$) of this triangle. Express the result as an integer or a fully simplified radical.

Simplify the following numerical expression fully. To do this, you must decompose the radicands into prime factors, extract factors outside the root and operate with fractions. The final result must be expressed using a single irreducible radical.

$$ A = 2\sqrt{75} - \sqrt{\frac{147}{4}} + \frac{5}{2}\sqrt{12} - \sqrt{300} $$

We have three water tanks with different cubic shapes.

• Tank A has a capacity of $2$ cubic feet.

• Tank B has an edge that measures exactly twice the edge of tank A.

• Tank C has an edge that measures half the edge of tank A.

• Calculate the exact length of the edge of tank A, expressing it as a radical.

• Calculate the exact volume of tank B and tank C using the properties of powers and radicals. Do not use decimals.

Express the result of the following operation as a single power with a fractional exponent. To achieve this, you must convert all radicals to powers, apply the properties of powers (product and quotient of equal bases) and simplify the final exponent.

$$ \frac{\sqrt[4]{a^3 \cdot \sqrt{a}}}{\sqrt{a \cdot \sqrt[3]{a^2}}} $$

Check if the irrational number $\phi = 1 + \sqrt{5}$ is a solution to the quadratic equation:

$$ x^2 - 2x - 4 = 0 $$

To do this, substitute $x$ with $(1 + \sqrt{5})$, expand the square of the binomial (algebraic identity) and operate the like terms until you see if the final result is zero.

The volume of a giant sphere used in a planetarium is $500 \pi$ cubic feet.

Knowing that the formula for the volume of a sphere is $V = \frac{4}{3}\pi r^3$:

1. Solve for the radius $r$ in terms of the volume.

2. Substitute the given volume and calculate the exact value of the radius by fully simplifying the radical (extracting factors outside the cube root).

Perform the following operations by grouping the terms into a single root. Remember that to multiply or divide radicals with a different index, you must first reduce them to a common index (least common multiple of the indices).

1. a) $ \sqrt{2} \cdot \sqrt[3]{4} \cdot \sqrt[6]{8} $

1. b) $ \frac{\sqrt[4]{x^3}}{\sqrt[6]{x}} $

A bacterium has a square shape and its area is $2,25 \cdot 10^{-10}$ square meters.

Calculate the side length of the bacterium. Express the final result in scientific notation and in millimeters. (Remember that $\sqrt{a \cdot 10^n} = \sqrt{a} \cdot \sqrt{10^n}$).