Fractional exponentsIB MYP Maths Extended: Revision notes

Section 1

Roots as powers

A fractional exponent with numerator 1 is a root: $a^{\frac1n}=\sqrt[n]{a}.$ So $a^{\frac12}$ is the square root, $a^{\frac13}$ is the cube root and $a^{\frac14}$ is the fourth root. For example $64^{\frac13}=4$ because $4^3=64$ , and $81^{\frac14}=3$ because $3^4=81$ . This works because $\left(a^{\frac1n}\right)^n=a^1=a$ , using the law $(a^m)^n=a^{mn}$ .

Key termsfractional exponentroot

Exam tip

Learn the common roots: squares to $15^2$ , cubes $2^3$ to $5^3$ and powers of 2 up to $2^6$ .

Section 2

Numerator and denominator

For a general fraction $\frac mn$ , the denominator is the root and the numerator is the power: $a^{\frac mn}=\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}.$ Take the root first, because it keeps the numbers small. Example: $8^{\frac23}=\left(\sqrt[3]{8}\right)^2=2^2=4$ . Another: $16^{\frac34}=2^3=8$ .

Key termsdenominatornumerator

Common mistake

Treating $a^{\frac23}$ as $a\times\frac23$ or as $\frac{a^2}{3}$ . The exponent is not a multiplier.

Section 3

Negative fractional exponents

A negative exponent means take the reciprocal: $a^{-\frac mn}=\frac{1}{a^{\frac mn}}.$ Example: $64^{-\frac23}=\frac{1}{64^{\frac23}}=\frac{1}{4^2}=\frac1{16}$ . For a fraction base, flip it: $\left(\frac{27}{8}\right)^{-\frac23}=\left(\frac{8}{27}\right)^{\frac23}=\left(\frac23\right)^2=\frac49$ . The sign of the exponent changes the position, not the sign of the answer.

Key termsreciprocal

Common mistake

Making the answer negative. $25^{-\frac12}=\frac15$ , not $-5$ .

Section 4

Laws of indices with fractions

The index laws work for fractions too: $a^m\times a^n=a^{m+n},\quad a^m\div a^n=a^{m-n},\quad (a^m)^n=a^{mn},\quad (ab)^n=a^nb^n.$ Use common denominators to add or subtract the indices: $x^{\frac23}\times x^{\frac16}=x^{\frac46+\frac16}=x^{\frac56}$ . Multiply indices for a power of a power: $\left(x^{\frac12}\right)^4=x^2$ .

Key termsindex laws

Common mistake

Adding numerators and denominators separately: $\frac23+\frac16\neq\frac39$ .

Section 5

Simplifying algebraic expressions

Apply the power to every factor of a bracket, numbers and letters. Example: $\left(16x^8\right)^{\frac34}=16^{\frac34}\times x^{8\times\frac34}=8x^6$ . Another: $\sqrt[3]{x}\div x^{\frac34}=x^{\frac13-\frac34}=x^{-\frac5{12}}$ . Write roots as fractional powers first, simplify, then convert back only if the question asks for a root.

Key termspower of a power

Exam tip

Number and letter parts are handled separately: simplify the number, then each letter.

Section 6

Using fractional exponents in context

Many real formulae use fractional exponents. A planet's period is $T=r^{\frac32}$ and an animal's energy use is $R=70m^{\frac34}$ . To evaluate, take the root of the base first. To rearrange, raise both sides to the reciprocal power: if $r^{\frac32}=8$ then $r=8^{\frac23}=4$ . Interpret answers in context and give units.

Key termsmodel

Exam tip

To undo a power $\frac mn$ , raise both sides to the power $\frac nm$ .

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Fractional exponents

Mia is exploring powers of $64$ without a calculator.
Evaluate $64^{-\frac{5}{6}}$ , giving your answer as a fraction.2 marks
Kofi is simplifying algebraic expressions in which $x>0$ .
Simplify $\sqrt[3]{x}\div x^{\frac34}$ , giving your answer as a single power of $x$ .2 marks
The orbital period $T$ (in Earth years) of a planet is modelled by $T=r^{\frac{3}{2}}$ , where $r$ is the planet's average distance from its star in astronomical units (AU).
A dwarf planet has $r=25$ . Find its orbital period $T$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).