Exercises: Extracting factors

To extract factors from a root, we group the repeated factors of the radicand as indicated by the index: in a square root (index $2$), each group of $2$ equal factors comes out as a single factor; in a cube root (index $3$), each group of $3$ comes out.

Look at the following diagram representing $\sqrt[3]{x^4}$. As $x^4 = x^3 \cdot x$, the group of $3$ comes out as a single $x$ and the remaining factor 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) $\sqrt[4]{y^5}$

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 comes out.

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

Work out the extraction of factors for these powers by breaking down the exponent mentally or by carrying out 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)

Many times we find numbers that are not factorised. 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 $2$ is squared, it can "escape" from the square root.

Decompose into prime factors and extract as much as 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 form (by extracting factors).

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

1. Factorise the number 72.

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

3. Extract the numbers that are squared.

Sometimes we have numbers and letters mixed inside a root. The procedure is the same: we work with the number on one side and with each letter 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) $\sqrt{25z^3}$

We can not only extract factors from the numerator, but also 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 top and bottom independently. In this example, $b^2$ comes out 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: factorise 27 and leave it in the denominator)

Let's play the inverse 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: $3\sqrt{2} = \sqrt{3^2 \cdot 2} = \sqrt{9 \cdot 2} = \sqrt{18}$.

In your exercise book, match each simplified expression on the left with its original root on the right (you don't need to calculate, just reason which number came 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. $\sqrt{18}$

Simplify the following radicals. Look closely at the index of the root, as this exercise mixes 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) $\sqrt[3]{5^4}$

To simplify the radical $\sqrt[3]{a^7 \cdot b^4}$, Lucía divides each exponent by the index: $7 : 3 = 2$ (remainder $1$) and $4 : 3 = 1$ (remainder $1$). Then, she writes the following step in her notebook:

$$\sqrt[3]{a^7 \cdot b^4} = a \cdot b \cdot \sqrt[3]{a^2 \cdot b}$$

1. a) Identify the conceptual error Lucía made when placing the quotients and the remainders of the divisions.
2. b) Using the power property $a^D = (a^c)^d \cdot a^r$, explain why the quotient must be the exponent of the outer factor and the remainder that of the inner factor.
3. c) Write the corrected simplification of this radical.

We want to analyze and approximate the radical $\sqrt{180}$ without initially using a calculator.

1. a) Estimate between which two consecutive integers $\sqrt{180}$ lies, justifying your answer based on the two closest perfect squares.
2. b) Factorize $180$ into prime factors and take all possible factors out of the root.
3. c) Knowing that $\sqrt{5} \approx 2,24$, check whether the value obtained by multiplying the outer coefficient by this approximation is consistent with the interval you estimated in part a).

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

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

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

In many scientific formulas, very large or very small powers of 10 appear inside roots. Simplify the following expressions by extracting powers of 10 from 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 $3 \cdot 10^{-5}$)

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

In order to add or subtract radicals, they must be like radicals. At first glance, the following radicals appear different. Extract factors from each of them to check whether their radical parts are the same and, if so, express them in simplified form:

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

Simplify the following fractions containing 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) $\sqrt[3]{\frac{16a^4}{27b^3}}$

State whether the following equalities are true or false by carrying out the necessary factor extraction:

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) $\sqrt{\frac{x^2}{4}} = \frac{x}{2}$

When solving a calculation challenge, you obtain the simplified radical $12\sqrt[3]{5}$ as your final result. The initial statement has been erased from the board, but you know it came from extracting factors from an exact cube root of the form $\sqrt[3]{A}$, where $A$ is a natural number.

1. a) Reconstruct the original radicand $A$, first expressing it through its prime factorization and then finding its numerical value.
2. b) If the simplified result had been $12\sqrt{5}$ (square root) coming from a radical of the form $\sqrt{B}$, which of the two original radicands ($A$ or $B$) is larger? Justify it by comparing how the index of the root influences taking factors in, without the need to calculate roots.

To simplify the radical $\sqrt{7200}$, two students propose different approaches:

• Carlos's approach: He factorizes $7200$ into prime factors ($7200 = 2^5 \cdot 3^2 \cdot 5^2$), divides the exponents by the index $2$, and takes each factor out according to its quotient and remainder.

• Marta's approach: She writes $7200$ by mentally associating known squares: $7200 = 72 \cdot 100 = 36 \cdot 2 \cdot 100 = 6^2 \cdot 10^2 \cdot 2$, and takes the bases directly outside.

1. a) Apply both methods detailing the intermediate steps and check that the same irreducible expression is obtained.
2. b) Evaluate both procedures: in which cases is Marta's approach clearly faster, and in which situations is it essential to resort to Carlos's algorithm?
3. c) If you are asked to simplify $\sqrt{1848}$, which of the two methods would you choose? Justify your choice.

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 real numbers. Extract all possible factors from the root.

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

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

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

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

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

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

In standard form, 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 standard form.

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

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.

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

The following polynomial inside the square root corresponds to a standard algebraic identity (the square of a binomial or trinomial). Factorise the polynomial inside the radicand and simplify the expression by extracting the resulting factors.

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

Calculate the volume of a cube knowing that its edge measures $a = \sqrt[3]{16x^4}$. Express the final result in simplified form, 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 express simplified $a$ first and then calculate the volume by operating with radicals.

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

In traffic accident investigation in Spain, the Directorate-General for Traffic (DGT) uses expressions derived from physics to calculate the estimated speed $v$ (in $\text{km/h}$) prior to emergency braking on a wet road surface:

$$v = \sqrt{250 \cdot \mu \cdot d}$$

where $\mu$ represents the tire-asphalt friction coefficient (dimensionless) and $d$ the braking distance (in meters). On a conventional road with a $90\text{ km/h}$ speed limit, with a wet surface where $\mu = \frac{2}{5}$, a skid mark of $d = 45\text{ m}$ is measured.

1. a) Substitute the data into the expression and simplify the radical by extracting all possible factors to express the speed in exact form as $k\sqrt{m}$, with $k, m \in \mathbb{N}$.
2. b) Knowing that $2,23 < \sqrt{5} < 2,24$, bound the value of the estimated speed and determine with justification whether the vehicle was exceeding the legal road speed limit before braking.
3. c) A witness states: "If the mark had been twice as long ($90\text{ m}$), it would mean that the vehicle was traveling at exactly twice the speed." Use factor extraction from the radical to algebraically justify whether the witness's statement is true or false.

In an exercise with algebraic variables, you are asked to extract factors from the radical $\sqrt{\frac{50x^3}{y^6}}$, specifying in the problem statement that $x > 0$ and $y < 0$. Sergio hands in the following answer:

$$\sqrt{\frac{50x^3}{y^6}} = \sqrt{\frac{2 \cdot 5^2 \cdot x^2 \cdot x}{(y^3)^2}} = \frac{5x}{y^3}\sqrt{2x}$$

1. a) Show that Sergio's result is contradictory by analyzing the signs: determine with reasoning what sign the initial radical has and what sign Sergio's final expression has according to the conditions of the statement ($x > 0$, $y < 0$).
2. b) Applying the property $\sqrt{a^2} = |a|$, correct Sergio's working to provide the correct simplified expression without using absolute value bars.
3. c) If the expression had an odd index, for example $\sqrt[3]{\frac{50x^4}{y^6}}$ with the same conditions for $x$ and $y$, would this same sign conflict have arisen when extracting factors? Justify your answer based on the properties of odd-index roots.