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) $2\sqrt[3]{16} - \sqrt[3]{54}$

Fully simplify the following numerical expression. To do this, prime factorise the radicands, take factors out of the root and work with fractions. The final answer must be expressed using a single irreducible radical.

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

Sometimes the numbers inside the root have factors left over that can come out. Extract all possible factors from the following radicals.

• $\sqrt{2^5}$

• $\sqrt{3^3}$

• $\sqrt{5^4}$

• $\sqrt[3]{2^4}$

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

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

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

• $\sqrt{2} \times \sqrt{32}$

• $\sqrt[3]{2} \times \sqrt[3]{4}$

• $\sqrt{5} \times \sqrt{20}$

• $\sqrt{3} \times \sqrt{27}$

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

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

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

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

When we have a root inside another root, we can simplify it by multiplying their indices ($\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \times m]{a}$). 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) $\sqrt{2 \times \sqrt{2}}$ (Hint: Move the first 2 inside the second square root)

Express the result of the following operation as a single power with a fractional exponent. To do this, convert all radicals to powers, apply the index laws (product and quotient of powers with the same base), and simplify the final exponent.

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

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

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

2. Find the total perimeter of the plot.

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 the Pythagorean theorem ($h = \sqrt{a^2 + b^2}$), calculate the length of the screen's diagonal in centimeters.

Note: The result is an exact root.

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

We want to attach a guy wire to support a vertical post 4 meters high. The wire will be anchored to the ground at a distance of $\sqrt{20}$ meters from the base of the post. Calculate the exact length of cable needed using the Pythagorean 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 $V = \text{arista}^3$)

An architectural firm has designed a decorative triangular piece for a facade. The piece is an isosceles triangle (two equal sides and one different side).

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

Using the Pythagorean theorem and the rules of radicals (taking factors out and simplifying), calculate the exact height ($h$) of the triangle. Express the answer as an integer or a fully simplified radical.

We have three water tanks with different cubic shapes.

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

• Tank B has an edge length that is exactly double that of Tank A.

• Tank C has an edge length that is half that of Tank A.

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

• Calculate the exact volume of Tank B and Tank C using laws of exponents and rules of radicals. Do not use decimals.

Check whether 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 collect like terms to see if the final result is zero.

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

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

1. Rearrange the formula to make the radius $r$ the subject in terms of the volume.

2. Substitute the given volume and calculate the exact value of the radius, fully simplifying the radical (taking factors out of the cube root).

A bacterium is square in shape and has an area of $2.25 \times 10^{-10}$ square meters.

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