Exercises: Powers with rational exponents. Radicals

In the study of microorganisms, the size of certain structures can be expressed using powers and roots. Convert the following radicals into powers with fractional exponents and vice versa:

1. a) $\sqrt[5]{x^2}$

1. b) $\frac{1}{\sqrt[3]{a}}$

1. c) $7^{\frac{3}{4}}$

1. d) $x^{-\frac{1}{2}}$

Simplify the following radicals by dividing the index and the exponent of the radicand by their highest common factor:

1. a) $\sqrt[12]{x^8}$

1. b) $\sqrt[9]{27}$

1. c) $\sqrt[10]{a^{15}}$

1. d) $\sqrt[6]{125}$

To optimise space in a graphic design, a programmer needs to extract factors from the radicals whenever possible. Extract factors from the radicals in the following expressions:

1. a) $\sqrt{50}$

1. b) $\sqrt[3]{16x^5}$

1. c) $\sqrt{a^3 b^4 c}$

1. d) $\sqrt[4]{32}$

In some financial formulas, it is preferable to have all elements under a single radical sign. Insert the factors inside the root and simplify the result:

1. a) $3\sqrt{2}$

1. b) $x^2 \sqrt[3]{x}$

1. c) $\frac{2}{3}\sqrt{5}$

1. d) $a \sqrt[4]{a^3}$

An architect is calculating the perimeters of a metal structure made up of several sections. Add and simplify the following like radicals to obtain the total length:

$2\sqrt{27} - 5\sqrt{3} + \sqrt{12}$

Perform the following multiplication and division operations with radicals having the same index, simplifying the final result:

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

1. b) $\frac{\sqrt{18}}{\sqrt{2}}$

1. c) $\sqrt[5]{x^2} \cdot \sqrt[5]{x^3}$

1. d) $(\sqrt[4]{a})^2$

An interior designer has planned a square rug for a law firm in Madrid. The total area of the rug is $18 \text{ m}^2$.

1. a) Express the length of the side of the square exactly using a radical.
2. b) Simplify the radical by extracting factors.

Rationalise the following expressions to eliminate the radicals from the denominator, simplifying the result where possible:

1. a) $\frac{5}{\sqrt{3}}$

1. b) $\frac{1}{\sqrt[3]{2}}$

1. c) $\frac{\sqrt{2}}{\sqrt{5}}$

1. d) $\frac{x}{\sqrt{x}}$

Marcos has to calculate the numerical value of the power $16^{-3/4}$ for a class exercise. In his notebook, he writes the following steps:

Step 1: $16^{-3/4} = -16^{3/4}$

Step 2: $-16^{3/4} = -(\sqrt[4]{16})^3$

Step 3: $-(\sqrt[4]{16})^3 = -(2)^3 = -8$

1. a) Point out, with reasoning, in which step Marcos makes the first mistake and explain what conceptual error he has regarding the meaning of the minus sign in the exponent.
2. b) Correct the exercise step by step: transform the power with a negative exponent into its equivalent expression with a positive exponent, write it in radical form, and calculate its exact value as an irreducible fraction.
3. c) Based on your result, explain the specific meaning of both the denominator 4 and the numerator 3 of the initial power applied to the number 16.

We want to analyze the numerical behavior of the powers $A = 75^{0,5}$ and $B = 75^{-0,5}$.

1. a) Without using a calculator, estimate between which two consecutive integers the value of $A = 75^{0,5}$ lies, justifying your reasoning based on the closest perfect squares.
2. b) Without calculating decimals, reason between which two integers $B = 75^{-0,5}$ is located and explain which simple fraction of the form $1/k$ (with $k$ an integer) its value approaches. Can a power with a positive base result in a negative number? Justify your answer.
3. c) Express both $A$ and $B$ in simplified radical form (extracting factors in $A$ and rationalizing the denominator in $B$). Knowing that $\sqrt{3} \approx 1,73$, check numerically whether your estimates from parts a) and b) have been consistent.

Express the following powers with rational exponents in radical form and simplify the result by taking out as many factors as possible:

1. a) $27^{\frac{2}{3}}$

1. b) $125^{-\frac{1}{3}}$

1. c) $\left(\frac{16}{81}\right)^{\frac{3}{4}}$

1. d) $x^{1,5}$

Calculate and simplify the following expression, leaving the result as a single radical with the smallest possible index:

$$\frac{\sqrt[3]{a^2} \cdot \sqrt[4]{a^3}}{\sqrt[6]{a^5}}$$

An architectural firm is designing a triangular garden. The lengths of the three paths bordering the garden are given by the following expressions in metres:

• Path A: $3\sqrt{20}$

• Path B: $\sqrt{125}$

• Path C: $2\sqrt{45}$

Calculate the total perimeter of the garden, simplifying the expression as far as possible.

Simplify the following expression by reducing to a single radical and taking out factors if necessary:

$$\sqrt{x \cdot \sqrt[3]{x^2 \cdot \sqrt{x}}}$$

Rationalise and simplify the following expressions so that they do not contain roots in the denominator:

1. a) $\frac{10}{\sqrt[3]{5^2}}$

1. b) $\frac{2\sqrt{3}}{\sqrt{7} - \sqrt{2}}$

1. c) $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$

The volume of a cubic container used to transport tech goods is $V = 16\sqrt{2}$ cubic metres.

1. a) Express the edge of the cube as a power with base 2.
2. b) Calculate the exact length of the edge in simplified radical form.

Solve the following combined operation with radicals, expressing the result in the simplest possible form:

$$\left( \sqrt{12} - \sqrt{3} \right) \cdot \sqrt[3]{9}$$

Hint: First express all terms as powers with base 3.

In an entrance exam, students are asked to determine which of the following two metal pieces has a greater area. Piece A is a square with side length $L_A = \sqrt[3]{4}$ cm and piece B is a square with side length $L_B = \sqrt[4]{6}$ cm.

Calculate the area of each piece and compare them by reducing the radicals to a common index.

A teacher asks their students to reconstruct an original radical expression $\sqrt[n]{M}$, where $n$ is a natural number ($n \ge 2$) and $M = c \cdot x^p \cdot y^q$ is a monomial with $c, p, q \in \mathbb{N}$, from the result they obtained when simplifying it.

This simplified result, expressed as a product of powers with irreducible rational exponents, is:

$$2 \cdot x^{5/6} \cdot y^{4/3}$$

1. a) Deduce the smallest common index $n$ necessary to write the entire product under a single radical of the form $\sqrt[n]{M}$, so that no fractional power appears in the radicand $M$.
2. b) Determine the explicit expression of the radicand $M = c \cdot x^p \cdot y^q$ (calculating the integer value of the coefficient $c$ and the exponents $p$ and $q$).
3. c) Starting from the single radical $\sqrt[n]{M}$ found, carry out the standard process of extracting factors and verify that the simplified expression is recovered. Clearly state how many factors of $y$ come out of the root and how many remain inside.

According to real estate market data based on the CPI (INE), the average price per square meter of rent in a certain Spanish city accumulated an increase of $33,1\,\%$ over a period of 3 consecutive years.

1. a) Model the price evolution considering a constant annual growth factor $k = (1 + r)$, where $r$ is the annual rate as a decimal. Write the relationship between the initial price $P_0$ and the final price $P_3$, and solve for $k$, expressing it in radical form and as a power with a fractional exponent.
2. b) Knowing that $1,331 = 1,1^3$, calculate the exact value of the annual rate of change $r$ and express it as a percentage.
3. c) A neighborhood representative claims in the press that rent has increased «by $11,03\,\%$ each year», since they divided $33,1\,\%$ by 3. Demonstrate using powers why this reasoning is erroneous and justify whether simple division overestimates or underestimates the actual annual rate.
4. d) If a rental contract increases at that same constant rate of $10\,\%$ per year ($k = 1,1$), by what factor will the price have been multiplied after a year and a half ($t = 1,5$ years)? Express this factor as a rational power, transform it into a simplified radical, and reason whether the increase after 18 months is greater than, less than, or equal to $15\,\%$.

Simplify the following algebraic expression by operating with the radicals and taking out as many factors as possible. Express the final result using a single radical:

$\displaystyle \frac{\sqrt{x^3 \cdot \sqrt[3]{x^2 \cdot y}}}{\sqrt[6]{x \cdot y^5}} \cdot \left( \frac{x}{y^2} \right)^{-1/2}$

Calculate the exact total surface area of a regular octahedron (formed by two square-based pyramids joined at their base) whose edge length is $\sqrt{6} + \sqrt{2}$ cm. You must leave the result simplified in radical form, without using decimal approximations.

(Hint: The area of an equilateral triangle of side length $a$ is $A = \frac{\sqrt{3}}{4}a^2$)

Rationalise and simplify the following expression as far as possible. Ensure that the final denominator is an integer:

$\displaystyle \frac{2\sqrt{3}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$

In the design of a new microelectronic component, its resistance is given by the expression:

$\displaystyle R = \sqrt[4]{27} \cdot \sqrt[3]{9} \cdot \sqrt{3}$

Express the radicals as powers of base 3 to find the value of $R$ as a single power with a fractional exponent. Afterwards, state which two consecutive integers this value lies between.

Find the value of $x$ that satisfies the following equation, simplifying all radicals beforehand:

$\displaystyle \sqrt{125x} - 2\sqrt{45x} + \sqrt{500x} = 18\sqrt{5}$

An architect is designing a right-angled triangular window. The hypotenuse measures $\sqrt[4]{1024}$ metres and one of the shorter sides measures $\sqrt{2}$ metres.

1. a) Calculate the exact length of the other side in simplified radical form.
2. b) Calculate the area of the window. Is the area a rational or irrational number? Justify your answer.

Simplify the following combined expression, leaving the result in its simplest form:

$\displaystyle \frac{\left( \sqrt[3]{a^2 \cdot b} \right)^2 \cdot \sqrt{a \cdot b^3}}{\sqrt[6]{a^5 \cdot b}}$

A cubic water tank is to be constructed with an interior diagonal measuring exactly $\sqrt{108}$ metres.

1. a) Show that the edge length of the tank is a rational number.
2. b) If the cost of waterproofing the interior walls (the 4 side faces and the floor) is $150 \cdot \sqrt{3}$ euros per square metre, calculate the exact total cost of the work.

To simplify the following expression with positive real variables:

$$E = \frac{\sqrt{x \cdot \sqrt[3]{x^2}}}{\sqrt[4]{x^3 \cdot \sqrt{x}}}$$

two 1st-year Bachillerato students propose two different resolution methods:

• Method A (Radicals and least common index): Introduce factors into the successive nested radicals, multiply the indices of consecutive roots, find the least common multiple of the indices of the numerator and denominator to unify into a single radical, and simplify the resulting fraction.

• Method B (Rational exponents): Immediately convert each root into a power with a fractional exponent, apply the properties of the product and quotient of powers with the same base by adding and subtracting fractions, and finally express the result as a simplified radical.

1. a) Solve the simplification of $E$ step by step following Method A, making each transformation on the radicals explicit.
2. b) Solve the simplification of $E$ step by step following Method B, showing in detail the arithmetic operations with the rational exponents.
3. c) Critically compare both methods: reason which of the two is more advantageous and reliable when the algebraic expression contains several different bases with negative exponents, such as in expressions of the type $\left(x^2 \cdot y^{-3}\right)^{-1/4}$.

In business economics, the Cobb-Douglas production function models the amount of output $Y$ that a company generates from the units of invested capital ($K$) and the hours of labor employed ($L$). For a transport and logistics company, the model is given by:

$$Y = 4 \cdot K^{0,75} \cdot L^{0,25} \quad (K > 0, \; L > 0)$$

1. a) Express the function $Y$ using a single radical with the smallest possible natural index, grouping the variables $K$ and $L$ in its radicand.
2. b) A function exhibits «constant returns to scale» if multiplying all production factors simultaneously by a factor $\lambda > 0$ results in the final output being multiplied by that exact same factor $\lambda$ (that is, $Y(\lambda K, \lambda L) = \lambda \cdot Y(K, L)$). Use the algebraic properties of powers with rational exponents to rigorously demonstrate whether or not this company exhibits constant returns to scale when it doubles its resources ($\lambda = 2$).
3. c) During a quarter, the company employs $K = 16$ units of capital and $L = 81$ hours of labor. Calculate the exact output $Y$ without using a calculator, showing the prime factor decomposition of the bases and the calculation with the rational exponents.
4. d) If the company's management decides to quadruple only the capital investment ($4K$) while keeping labor $L$ constant, by what factor is the total output $Y$ multiplied? Express it as a power with base 2 and in simplified radical form, and reason why this increase does not reach a fourfold increase in final output.