Simplifying Radicals

Often we find numbers that are not factored. The first step must always be to decompose the number into prime factors.

For example, to simplify $\sqrt{12}$, we first think that $12 = 2^2 \cdot 3$. As the $2$ is squared, it can "escape" from the square root.

Decompose into prime factors and extract everything you can from these square roots:

1. a) $\sqrt{18}$
2. b) $\sqrt{50}$
3. c) $\sqrt{20}$
4. d) $\sqrt{75}$

We have a square plot of land. We know that its area is $72 \text{ m}^2$, but we want to know exactly how long its side is, expressed in the simplest possible way (extracting factors).

Remember that if the Area of a square is $l^2$, then the side is $l = \sqrt{\text{Area}}$.

1. Factor the number 72.

2. Write $\sqrt{72}$ using those factors.

3. Extract the numbers that are squared.

To extract factors from a radical, we divide the exponent of the radicand by the index of the root.

• The quotient is the exponent of the factor that goes outside.

• The remainder is the exponent of the factor that stays inside.

Calculate the extraction of factors from these powers by mentally decomposing the exponent or performing the division:

1. a) $\sqrt{5^7}$ (Divide 7 by 2)
2. b) $\sqrt[3]{2^8}$ (Divide 8 by 3)
3. c) $\sqrt[4]{3^9}$ (Divide 9 by 4)

Simplify the following radicals. Look closely at the index of the root, as in this exercise we mix square roots (index 2) and cube roots (index 3).

Remember:

• In $\sqrt{\quad}$, we make groups of 2.

• In $\sqrt[3]{\quad}$, we make groups of 3.

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

Extract all possible factors from the following numerical radicals. To do this, you must first perform the prime factorization of the numbers inside the root.

1. a) $\sqrt{216}$
2. b) $\sqrt[3]{5000}$
3. c) $\sqrt[4]{162}$

Calculate and simplify the following radical by decomposing the radicand into prime factors. Be careful, as the radicand is a negative number and the index is odd, so the result exists in the real numbers. Extract all possible factors outside the root.

$$ \sqrt[3]{-54000} $$

Imagine that exponents are like groups of friends. The index of the root tells you how many of them have to group together to be able to leave the "house" (the radical). If it is a square root (index 2), they leave in pairs. If it is a cube root (index 3), they leave in threes.

Look at the following diagram representing $\sqrt[3]{x^4}$. There is a group of 3 that can go outside as a single $x$, and one is left over which stays inside.

Following this logic, extract all possible factors from the following roots:

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

Sometimes we have numbers and variables mixed inside a root. The procedure is the same: we work with the number on one hand and with each variable on the other.

Simplify the following radicals by extracting all possible factors:

1. a) $\sqrt{8x^3}$
2. b) $\sqrt{27a^2}$
3. c) $\sqrt{16y^5}$
4. d) $\sqrt{25z^3}$

Simplify the following radicals containing variables. Remember that to be able to extract a variable, its exponent must be equal to or greater than the index of the root.

1. a) $\sqrt[3]{x^4 \cdot y^7 \cdot z^{10}}$
2. b) $\sqrt{25a^3 b^5 c}$
3. c) $\sqrt[5]{\frac{m^7}{n^{10}}}$

We can not only extract factors from the numerator, we can also do it from the denominator if we have fractions inside the root.

If we have $\sqrt{\frac{a^3}{b^2}}$, we can take factors out from the top and bottom independently. In this example, $b^2$ goes outside as $b$ in the denominator, and from $a^3$ we take out an $a$ in the numerator: $\frac{a}{b}\sqrt{a}$.

Try to simplify these expressions:

1. a) $\sqrt{\frac{50}{9}}$
2. b) $\sqrt{\frac{x^3}{4}}$
3. c) $\sqrt{\frac{1}{27}}$ (Hint: factor 27 and leave it in the denominator)

Simplify the following fractions with radicals by extracting factors from both the numerator and the denominator. If possible, simplify the resulting fraction:

1. a) $\sqrt{\frac{98}{75}}$
2. b) $\sqrt[3]{\frac{16a^4}{27b^3}}$

Simplify the following algebraic expression with radicals. To do this, you must use the properties of powers to group similar terms inside the root before extracting the factors. Remember that a negative exponent is equivalent to the reciprocal of the base with a positive exponent.

$$ \sqrt[4]{\frac{32 \cdot x^6 \cdot y^{-3}}{81 \cdot z^5}} $$

State whether the following equalities are true or false by performing the necessary extraction of factors:

1. a) $\sqrt{a^2 + b^2} = a + b$
2. b) $\sqrt{16 \cdot x^6} = 4x^3$
3. c) $\sqrt[3]{8 + 27} = 5$
4. d) $\sqrt{\frac{x^2}{4}} = \frac{x}{2}$

Sometimes, before extracting factors, it is necessary to calculate inside the radicand. In this case, you cannot extract factors directly because there is an addition and a subtraction. First, factor out a common factor inside the root and, then, extract everything possible outside.

$$ \sqrt{18x^5 + 50x^5 - 8x^5} $$

The following polynomial inside the square root corresponds to a special product (the square of a binomial or trinomial). Factor the polynomial inside the radicand and simplify the expression by extracting the resulting factors.

$$ \sqrt{9x^2 - 30x + 25} $$

In many scientific formulas, very large or very small powers of 10 appear inside roots. Simplify the following expressions by extracting powers of 10 outside the radical:

1. a) $\sqrt{1,44 \cdot 10^7}$
2. b) $\sqrt[3]{2 \cdot 10^{14}}$
3. c) $\sqrt{0,00003}$ (Hint: express it first as $3 \cdot 10^{-5}$)

In scientific notation, to extract factors from a root, we sometimes need to adjust the exponent of the power of 10. Calculate the exact value of the following square root, extracting all possible factors and leaving the result expressed in simplified scientific notation.

$$ \sqrt{1,2 \cdot 10^7} $$

Simplify the following nested radical. First, convert the expression into a single root by multiplying the indices and express the radicand as a power. Finally, extract all possible factors.

$$ \sqrt{\sqrt[3]{64 \cdot a^{13} \cdot b^7}} $$

Let's play the reverse process to check if you have understood extraction. If I have a factor outside the root and I want to put it inside, I have to raise it to the index of the root.

For example: $3\sqrt{2} = \sqrt{3^2 \cdot 2} = \sqrt{9 \cdot 2} = \sqrt{18}$.

Match in your notebook each simplified expression on the left with its original root on the right (you do not need to calculate, just reason which number has gone out):

1. $2\sqrt{2}$ --- A. $\sqrt{12}$

2. $3\sqrt{2}$ --- B. $\sqrt{20}$

3. $2\sqrt{3}$ --- C. $\sqrt{8}$

4. $2\sqrt{5}$ --- D. $\sqrt{18}$

Calculate the volume of a cube knowing that its edge measures $a = \sqrt[3]{16x^4}$. Express the final simplified result by extracting all possible factors from the resulting root. Remember that the volume of a cube is $V = a^3$, but in this exercise we want you to first express $a$ simplified and then calculate the volume by calculating with radicals.

$$ \text{Calcula } V \text{ if } a = \sqrt[3]{16x^4} $$

To be able to add or subtract radicals, they must be similar. At first glance, the following radicals look different. Extract factors from each of them to check if their radical part is the same and, if so, express them in a simplified form:

1. a) $\sqrt{80}$
2. b) $\sqrt{125}$
3. c) $2\sqrt{45}$

We have a cubic water tank. We know that its total volume is $686 \text{ m}^3$. Calculate the edge length of the tank, expressing the result as a simplified radical (extracting factors).

Calculate the length of the hypotenuse of a right triangle whose legs measure $10 \text{ cm}$ and $5 \text{ cm}$. Apply the Pythagorean theorem and express the final result by extracting all possible factors from the root.

We have a square plot of land whose area is $720 \text{ m}^2$. We want to install an irrigation pipe that crosses the plot diagonally. Calculate the exact length of the pipe, expressing the result as a simplified radical (extracting the maximum number of factors).