Exercises: Powers with rational exponents: Radicals

Express the following powers with fractional exponents as radicals and calculate their exact value if it is an integer:

• a) $27^{\frac{1}{3}}$

• b) $16^{\frac{3}{4}}$

• c) $5^{\frac{2}{3}}$

Write the following radicals in the form of a power with a fractional exponent and simplify the fraction in the exponent:

• a) $\sqrt[6]{x^2}$

• b) $\sqrt[15]{a^5}$

• c) $\sqrt[8]{2^6}$

Take out all possible factors from inside the following radicals:

• a) $\sqrt{50}$

• b) $\sqrt[3]{16}$

• c) $\sqrt[4]{x^7 y^5}$

Insert the factors into the radical and simplify the resulting expression if possible:

• a) $3 \sqrt{2}$

• b) $x^2 \sqrt[3]{x}$

• c) $2a \sqrt[4]{3a^2}$, with $a \ge 0$

Calculate the following products of radicals of the same index and take out factors from the result if possible:

• a) $\sqrt{3} \cdot \sqrt{12}$

• b) $\sqrt[3]{4} \cdot \sqrt[3]{10}$

Calculate and simplify the result of the following divisions of radicals:

• a) $\frac{\sqrt{72}}{\sqrt{2}}$

• b) $\frac{\sqrt[3]{40}}{\sqrt[3]{5}}$

Calculate the following powers of radicals, moving the exponent inside the radicand and expressing the result in the simplest form possible:

• a) $(\sqrt[3]{5})^6$

• b) $(\sqrt{x^3})^4$

Express as a single radical using the root of a root rule and simplify the index if possible:

• a) $\sqrt{\sqrt[3]{64}}$

• b) $\sqrt[4]{\sqrt{x^8}}$

Elena has to calculate the exact value of the power $32^{-2/5}$. In her notebook, she writes the following solution:

At that point, she stops and claims that the result cannot be calculated by hand because $32^5$ is too large a number.

1. a) Find and explain with reasoning the two conceptual errors that Elena made when applying the properties of exponents and radicals.
2. b) Show the correct step-by-step development to obtain the numerical value as an irreducible fraction.
3. c) Justify why, when calculating an expression of the form $a^{m/n}$ manually, it is much more convenient to apply the root first and then the power, rather than doing it in the reverse order.

You want to estimate the numerical value of the rational exponent power $50^{3/2}$ without using a calculator.

1. a) Make a reasoned preliminary estimation: write $50^{3/2}$ in radical form and notice that the number $50$ is very close to the perfect square $49$. What integer should the final result be very close to? Justify your conjecture.
2. b) Write $50^{3/2}$ as a radical of the form $\sqrt{50^3}$, decompose the radicand into prime factors, and extract all possible factors out of the root.
3. c) Taking the approximation $\sqrt{2} \approx 1,414$, calculate the approximate decimal value from the simplified expression of the previous section and check whether your initial estimation from section a) was consistent by calculating the difference between both values.

In a biology laboratory, the number of bacteria in a given culture follows the expression $N(t) = 5000 \cdot 8^{\frac{t}{3}}$, where $t$ is the time elapsed in hours. Express this formula using radicals and calculate the exact number of bacteria present after $2$ hours.

For the manufacture of three electronic components, protective sheets are required whose thicknesses, measured in millimetres, are given by the expressions $A = \sqrt[3]{5}$, $B = \sqrt[4]{7}$ and $C = \sqrt[6]{24}$. Express the radicals with a common index to order the sheet thicknesses from smallest to largest.

Perform the following operations, first expressing the radicals with a common index, and take out any possible factors from the final result:

• a) $\sqrt{2} \cdot \sqrt[3]{4} \cdot \sqrt[4]{8}$

• b) $\frac{\sqrt[4]{27}}{\sqrt[6]{3}}$

A landscape architect is designing a green space in the shape of a scalene triangle and needs to calculate its perimetre to order fencing. The plans indicate that the sides measure $\sqrt{48}$ m, $\sqrt{75}$ m and $\sqrt{12}$ m. Simplify the radicals by taking out factors and calculate the exact perimetre expressed as a single radical.

A circular town square has an area of $120\pi$ m$^2$. Given that the area of a circle is given by the formula $A = \pi r^2$, calculate the exact radius of the square, fully simplifying the radical.

Calculate using the laws of fractional exponents and express the final result as a simplified radical:

An agricultural cooperative wants to fence a square piece of land whose total area is $1250 \text{ m}^2$. Calculate the exact length of the side of the land by taking out factors from the radical and determine how many metres of fencing they will need in total, leaving the result in its simplest radical form.

Simplify the following expression with nested radicals by progressively moving factors into the innermost radicals and expressing the final result as a single root:

To simplify the radical expression $E = \frac{\sqrt[3]{16} \cdot \sqrt[4]{8}}{\sqrt[6]{32}}$, two solving strategies can be proposed:

• Method 1 (rational exponent): Express each radicand in base $2$, convert all roots into fractional powers, and operate by applying the properties of the product and quotient of powers with the same base.

• Method 2 (common index): Express each radicand in base $2$, reduce all radicals to the least common index, and operate inside a single root to then simplify and extract factors.

1. a) Apply Method 1 detailing each step and write the final result as an irreducible radical with extracted factors.
2. b) Apply Method 2 detailing each step and verify that exactly the same result is obtained.
3. c) Compare both procedures: what mathematical relationship exists between the least common multiple of the denominators in the fractional powers and the common index of the radicals? Which of the two methods seemed more direct to you and why?

For the digital processing of a positive electrical signal $a$, a telecommunications team needs to combine two filters represented by radicals such that the following equality holds:

where the filter indices, $m$ and $n$, must be strictly positive integers satisfying the condition $m < n$.

1. a) Write the left-hand side as a single power of base $a$ and fractional exponent in terms of $m$ and $n$.
2. b) From the equality of exponents, deduce with reasoning what are the only two possible pairs of integers $(m, n)$ that satisfy the condition with $m < n$. Justify why there cannot be any other combination.
3. c) For each of the two pairs found, verify the equality by reducing the radicals on the left-hand side to a common index.

The space diagonal of a cubic water storage tank (the straight line joining two opposite vertices passing through the centre of the cube) measures exactly $\sqrt{108}$ metres. Calculate the exact length of the edge of the cube and, from it, determine the capacity of the tank in cubic metres. Express all results simplifying the radicals.

Fully simplify the following algebraic expression, assuming that $a$ and $b$ are positive real numbers. Express the final result using radical notation, leaving no negative or fractional exponents in the solution:

Kepler's third law relates the orbital period $T$ of a planet to the mean radius of its orbit $r$ using the expression $T = k^{\frac{1}{2}} \cdot r^{\frac{3}{2}}$. If the constant for a fictional star system is $k = 4 \cdot 10^{-6}$ and a planet orbits at a mean radius of $r = 25 \cdot 10^4$ km, calculate the exact orbital period using the properties of powers with rational exponents and radicals.

Find the value of $x$ that satisfies the following equation, first transforming all expressions into powers of base $2$ with rational exponents:

Express the final solution as an irreducible fraction.

The final amount $C_f$ obtained by investing an initial principal $C_0$ at an annual compound interest rate $r$ (expressed as a decimal) for $t$ years is given by the formula $C_f = C_0 (1+r)^t$. A company invests $10000$ € at an interest rate of $21$ % per year ($r = 0,21$). Using powers with fractional exponents, calculate the exact accumulated amount after $1$ year and $4$ months (which is equivalent to $t = \frac{4}{3}$ of a year). Knowing that $1,21 = 1,1^2$, simplify the result by taking out as many factors as possible from the radical.

Prove, using strictly the properties of powers with rational exponents, that for any positive real number $x$, the expression $\sqrt[n]{\sqrt[m]{x}}$ is equivalent to $\sqrt[m]{\sqrt[n]{x}}$. Then, apply this property or conversion to powers to calculate the exact numerical value of the following sum:

Simplify the following expression with nested radicals, reducing it to a single common index and taking out as many factors as possible from the resulting radical:

In maritime navigation, the distance $d$ (in kilometres) to the visible horizon for an observer situated at a height $h$ (in metres) above sea level is approximated by the formula $d = \sqrt{13h}$. Two observers are at different points on a cliff at heights of $h_1 = 52$ m and $h_2 = 117$ m respectively. Calculate the exact difference between their distances to the visible horizon, simplifying the expression $\sqrt{13 \cdot 117} - \sqrt{13 \cdot 52}$ by taking out factors.

In wildlife monitoring programmes at Doñana National Park, the veterinary team uses Kleiber's allometric law to estimate the daily basal metabolic rate $B(M)$ (energy consumed at rest, in kilocalories per day) of a mammal as a function of its body mass $M$ (in kilograms):

1. a) Express the formula for $B(M)$ using radicals and calculate the exact daily energy consumption of an adult Iberian lynx weighing $16\text{ kg}$ and that of a hind weighing $81\text{ kg}$.
2. b) The hind has a mass approximately $5$ times greater than that of the lynx ($81/16 \approx 5,06$). Prove whether its daily energy expenditure is also multiplied by $5$ by calculating the exact quotient $\frac{B(81)}{B(16)}$ using properties of fractional powers.
3. c) The specific metabolic rate per unit of body mass is defined as $b(M) = \frac{B(M)}{M}$. Prove algebraically, by applying properties of powers with rational exponents, that $b(M) = \frac{70}{\sqrt[4]{M}}$. From this formula, justify why the smallest mammals of the Iberian fauna need to ingest proportionally much more food per kilo of their own body than large herbivores.

In an automotive factory in Martorell, the cutting time $T$ (in seconds) used by a laser robot on a sheet metal of thickness $x$ (in millimetres, with $x > 0$) is modelled by the expression:

1. a) Successively convert each radical into a power with a rational exponent and simplify the complete expression step by step until obtaining a single power of the form $T(x) = x^k$, indicating the value of the irreducible fraction $k$.
2. b) For a high-strength structural part, the robot took exactly $8$ seconds to make the cut. Set up the equation $T(x) = 8$ and calculate the exact thickness $x$ of the sheet metal by solving for it using rational exponents. Express the final result by extracting all possible factors from an irreducible radical.
3. c) A plant technician claims that if the thickness of the sheet metal is doubled ($2x$), the cutting time will also double. Formally prove whether their claim is true or false by calculating the ratio $\frac{T(2x)}{T(x)}$ and expressing it as an irreducible radical.