1.10 Rational exponents

Learning objectives

• State the meaning of $a^{\frac{1}{n}}$ and $a^{\frac{m}{n}}$ as roots and powers, and write down the value of simple rational powers without a calculator.

• Calculate the value of an expression with a rational exponent, exactly by hand and with a GDC.

• Simplify numerical expressions with rational exponents using the laws of exponents.

• Simplify algebraic expressions with rational and negative rational exponents.

• Simplify surds and rationalise the denominator of a fraction.

• Express an expression involving roots and fractions in the form $px^{q}$, or as a sum of such terms.

• Solve equations of the form $x^{\frac{m}{n}} = k$ and rearrange formulae that contain rational powers.

• Interpret a formula with a rational exponent in a real context, such as Kleiber's law or Kepler's third law.

Introduction

Integer exponents count repeated multiplication: $2^{5}$ means five 2s multiplied together. Many formulas in science and economics, however, contain powers such as $x^{\frac{1}{2}}$, $w^{\frac{3}{4}}$

Rational exponents appear wherever one quantity grows in proportion to a fractional power of another. An animal's energy needs grow roughly with the $\frac{3}{4}$ power of its mass, a planet's orbital period grows with the $\frac{3}{2}$ power of its distance from the Sun, and the surface area of similar objects grows with the $\frac{2}{3}$

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Roots

Square roots and cube roots

The square root of a positive number $a$ is the positive number whose square is $a$. Since $7^{2} = 49$, the square root of 49 is 7, written $\sqrt{49} = 7$

Cube roots undo cubing. Since $4^{3} = 64$, the cube root of 64 is 4, written $\sqrt[3]{64} = 4$

The nth root

Square roots and cube roots are the first two members of a family. The fourth root of 81 is 3, because $3^{4} = 81$; the fifth root of 32 is 2, because $2^{5} = 32$.

The small number $n$ is the index of the root. Odd roots exist for every real number and keep its sign, while even roots exist only for numbers that are zero or positive. The table lists the facts most often used, and knowing them by heart makes exact evaluation much faster.

Some numbers appear in more than one row: $64 = 8^{2} = 4^{3} = 2^{6}$, so $\sqrt{64} = 8$

Exponents of the form 1/n

Why a^(1/2) means the square root

The laws of exponents from subtopic 1.5 were built for whole-number exponents. One of them states that when powers of the same base are multiplied, the exponents are added: $a^{m} \times a^{n} = a^{m+n}$. If this law is to keep working for fractions, a power with exponent $\frac{1}{2}$

So $a^{\frac{1}{2}}$ is a number which, multiplied by itself, gives $a$. That is exactly the square root, so for $a \ge 0$

The power-of-a-power law, $\left(a^{m}\right)^{n} = a^{mn}$, gives the same result: $\left(a^{\frac{1}{n}}\right)^{n} = a^{\frac{1}{n} \times n} = a^{1} = a$

Exponents of the form m/n

Two equivalent orders

A general rational exponent $\frac{m}{n}$ can be split as $\frac{1}{n} \times m$

In words, the denominator $n$ is the root and the numerator $m$ is the power. Both orders give the same answer, so the choice is about convenience. Taking the root first keeps the numbers small; raising to the power first produces large intermediate values that are harder to root.

For $32^{\frac{3}{5}}$, taking the fifth root first gives $\sqrt[5]{32} = 2$

A fraction is rooted by rooting its numerator and its denominator separately, as the last row shows. Decimals are often easier as fractions: $0.04 = \frac{4}{100} = \frac{1}{25}$, so $0.04^{\frac{3}{2}} = \left(\frac{1}{5}\right)^{3} = \frac{1}{125} = 0.008$

Exponents written as decimals

Formulas often give rational exponents as decimals: $0.5 = \frac{1}{2}$, $0.25 = \frac{1}{4}$

Using the GDC

Any rational power can be found on a GDC with the power key (^ on most models). The exponent must be entered in brackets, because the calculator follows the order of operations exactly as written.

• 32^(3/5) gives 8, the intended value.

• 32^3/5 gives 6553.6, because the GDC calculates $32^{3}$ first and then divides by 5.

Most GDCs also have an $n$th-root template, $\sqrt[x]{\phantom{a}}$, in their maths or template menu, which is useful for roots of high index. When an exact value is needed, the fraction-conversion function turns a decimal answer such as 0.444444… into $\frac{4}{9}$

Negative rational exponents

Combining a reciprocal with a root

A negative exponent means a reciprocal: $a^{-n} = \frac{1}{a^{n}}$ (subtopic 1.5, Exponents and introduction to logarithms). The same rule holds for fractional exponents, so a negative rational exponent combines three operations: a reciprocal (from the minus sign), a root (from the denominator) and a power (from the numerator).

The three operations can be done in any order, so the order is chosen to keep the numbers simple. For a fraction, taking the reciprocal first turns it upside down: $\left(\frac{a}{b}\right)^{-k} = \left(\frac{b}{a}\right)^{k}$

A negative exponent never makes the value negative. For a positive base, every power, whatever its exponent, is positive. The negative sign only decides whether the base goes on the top or the bottom of a fraction.

The laws of exponents with rational exponents

The same laws, extended

Because rational exponents were defined to keep the laws of exponents true, every law from subtopic 1.5 can be used unchanged with fractions. For a positive base $a$ (and positive $b$) and any rational numbers $m$ and $n$:

Adding and subtracting fractional exponents needs a common denominator: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Why the base must be positive

The rules in the table hold for every positive base. With a negative base they can give contradictions. The exponent $\frac{2}{6}$ equals $\frac{1}{3}$, yet

Two routes that the laws treat as equal give different answers, so the laws cannot be trusted for negative bases. For this reason expressions with rational exponents are simplified on the assumption that the base is positive ($x > 0$), which is also true in almost every real context: lengths, masses, volumes and times are positive. Single odd roots of negative numbers, such as $\sqrt[3]{-8} = -2$

Simplifying numerical expressions

A numerical expression is simplified by writing every number as a power of the same small base, then using the laws on the exponents. Prime factors show the base to choose: $8 = 2^{3}$, $16 = 2^{4}$, $9 = 3^{2}$

Simplifying algebraic expressions

An algebraic expression is simplified in the same way. Roots are first written as powers, numbers and letters are dealt with separately, and the exponents of each letter are combined. The power-of-a-product law applies the exponent to every factor in the bracket, including the number.

Surds

Exact roots and surds

A root such as $\sqrt{16}$ is a whole number, but most roots are not. The value of $\sqrt{2}$

Surds are rational exponents in another notation: $\sqrt{5} = 5^{\frac{1}{2}}$

Simplifying a surd

A surd is in its simplest form when the number under the root has no square factor other than 1. To simplify, the largest square factor is split off and its root taken outside.

Using a smaller square factor first, such as $\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}$

Surds behave like algebraic terms. Like surds (the same number under the root) can be added and subtracted, just as $5x + 3x = 8x$, while unlike surds cannot be combined. Multiplication uses $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

Rationalising the denominator

Removing a root from the bottom of a fraction

An answer such as $\frac{6}{\sqrt{3}}$ is exact but awkward: dividing by an irrational number is hard to do by hand, and two equal answers may look different. Multiplying the top and bottom of the fraction by the same surd leaves its value unchanged (it is a multiplication by 1) but makes the denominator rational, because $\sqrt{a} \times \sqrt{a} = a$

The answer $2\sqrt{3}$ is simpler than $\frac{6}{\sqrt{3}}$

Denominators of the form a + √b

When the denominator is a sum or difference such as $3 - \sqrt{5}$, multiplying by $\sqrt{5}$

Writing expressions in the form px^q

Why the form matters

Many later methods work on single powers of $x$. The rule for differentiating in subtopic 5.9 and the rule for integrating in subtopic 5.11 both act on terms of the form $px^{q}$, and a power model $y = ax^{b}$ (a power law) is written the same way. Expressions that contain roots, fractions or brackets must first be rewritten as one power of $x$

The method

1. Write every root as a power: $\sqrt{x} = x^{\frac{1}{2}}$

The constant and the power are handled separately because they behave differently: a constant 5 in the denominator becomes $\frac{1}{5}$ in front, not $5^{-1}$ attached to $x$. Keeping $p$

Sums in the numerator

A fraction with a sum on top splits into separate fractions, one for each term, because $\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$

Formulas from science in the form px^q

The same rewriting turns physical formulas into a form ready for calculus or for comparing powers. The period of a simple pendulum of length $L$ metres is $T = 2\pi\sqrt{\frac{L}{g}}$

In the form $T = pL^{q}$, $p \approx 2.01$ and $q = \frac{1}{2}$

Solving equations with rational exponents

Undoing a rational power

An equation such as $x^{\frac{3}{2}} = 27$ is solved by raising both sides to the reciprocal power, $\frac{2}{3}$

The same idea solves $x^{\frac{m}{n}} = k$ in general: $x = k^{\frac{n}{m}}$

Kepler's third law

For planets orbiting the Sun, the square of the orbital period is proportional to the cube of the semi-major axis of the orbit (its average distance from the Sun). Measuring the period $T$ in Earth years and the distance $a$ in astronomical units makes the constant equal to 1, because the Earth has $T = 1$ and $a = 1$. So $T^{2} = a^{3}$

Mars orbits at about 1.524 AU, so its period is $1.524^{\frac{3}{2}} \approx 1.88$ years. The observed period is about 687 days, or $687 \div 365.25 \approx 1.88$

Source: Wikipedia, "Kepler's laws of planetary motion" and "Mars" (semi-major axis 1.524 AU, orbital period 687 days), 2026.

Part (c) shows a general feature of power formulas: multiplying the input by $k$ multiplies the output by $k^{q}$, independent of the starting value. Formulas of the form $y = px^{q}$ therefore describe scaling very directly.

Graphs of rational powers

The graph of $y = x^{q}$ for $x > 0$ has a shape set by the exponent. The graph of the three functions $y = x^{\frac{1}{2}}$

• For $0 < q < 1$, such as $q = \frac{1}{2}$ or $q = \frac{3}{4}$

Between 0 and 1 the order is reversed: $x^{\frac{3}{2}} < x^{\frac{1}{2}}$

Case study: Kleiber's law

An animal's basal metabolic rate (BMR) is the energy it uses each day at rest, just to stay alive. Larger animals use more energy in total, but not in proportion to their mass. Kleiber's law, an empirical rule found by comparing many species, states that

where $w$ is the animal's mass in kilograms. The exponent $\frac{3}{4}$ lies between 0 and 1, so the energy needed rises more slowly than the mass.

Source: Wikipedia, "Kleiber's law" (BMR ≈ 70w^(3/4) kcal per day), 2026.

Exact values

Masses that are perfect fourth powers give exact values, since $w^{\frac{3}{4}} = \left(\sqrt[4]{w}\right)^{3}$

The table uses masses chosen to give exact values (they are not data for particular species). The last column, the energy per kilogram of body mass, is $\frac{70w^{\frac{3}{4}}}{w} = 70w^{-\frac{1}{4}}$

The graph of Kleiber's law against the line $\text{BMR} = 70w$ (energy proportional to mass) shows how far apart the two models drift as mass grows.

Interpreting the exponent

The exponent gives the effect of scaling directly. Doubling the mass multiplies the BMR by $2^{\frac{3}{4}} \approx 1.68$, an increase of 68%, not 100%. Multiplying the mass by 16 multiplies the BMR by $16^{\frac{3}{4}} = 8$

Limitations of the model

Kleiber's law is an approximation that fits a broad trend across species and does not give the exact rate for any one animal. Individuals of the same mass differ with age, activity, temperature and body composition, and later studies have debated whether the best exponent is exactly $\frac{3}{4}$ or closer to $\frac{2}{3}$. The coefficient 70 also depends on the units chosen: in watts the same law reads

Scaling of similar solids

The same reasoning applies to the surface area $A$ and volume $V$ of similar solids. A cube of side $s$ has $V = s^{3}$ and $A = 6s^{2}$

A cube of volume $27\text{ cm}^3$ has $A = 6 \times 27^{\frac{2}{3}} = 6 \times 9 = 54\text{ cm}^2$

Exam focus

Rational exponents rarely make a whole question. They appear on HL Papers 1 and 2 as 1–3-mark parts that ask to simplify, express an expression "in the form $px^{q}$" or "in the form $ax^{n}$", or find an exact value, and they sit inside calculus questions (rewriting $\frac{3}{\sqrt{x}}$

Two features of this question are typical. The "show that" parts require each law of exponents to be applied in a visible line, working forwards from the given formulas, and the "exact" in part (d) rules out a decimal from the GDC.

Calculus questions in subtopics 5.9 and 5.11 often begin with a function such as $f(x) = \frac{6}{\sqrt{x}}$

Practice questions

1. Write down the value of: (a) $121^{\frac{1}{2}}$; (b) $216^{\frac{1}{3}}$

Summary

• $a^{\frac{1}{n}} = \sqrt[n]{a}$

Self-check

• To do:

  I can explain why $a^{\frac{1}{2}}$ must equal $\sqrt{a}$