HL · Logs & series 1 · Rational exponents: roots as powers

Module 10 of 47Lesson 1 of 9 · HL · Logs & series 1 · Rational exponents: roots as powers

Laws of logarithms, rational exponents, infinite series & log scales · Essential understanding 1

Rational exponents: roots as powers

IB AI AHL 1.10with Prof. Kitso

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

The big idea: A power does not have to be a whole number. The exponent $\dfrac{1}{2}$ means square root,

\dfrac{1}{3}

means cube root, and $\dfrac{m}{n}$ means "take the n-th root, then raise to the power m". The best part: every law of exponents you already know still works.

By the end of this lesson you can:

write roots as powers: $\sqrt[n]{a} = a^{\frac{1}{n}}$
evaluate expressions such as $8^{\frac{2}{3}}$ and $27^{-\frac{2}{3}}$ without a calculator
simplify algebraic expressions with fractional powers and write them as k xⁿ
solve equations such as x^{$\dfrac{3}{2}$} = 27 by raising both sides to a reciprocal power
use rational exponents in real power-law models

🧑🏾 Imagine this…

Think about

9^{\frac{1}{2}}

. Use the power law

(a^m)^n = a^{mn}

: if you square it you get

\left(9^{\frac{1}{2}}\right)^2 = 9^{\frac{1}{2} \times 2} = 9^1 = 9

. So

9^{\frac{1}{2}}

is a number which, when squared, gives 9. That is exactly what

\sqrt{9}

means! The same idea with

8^{\frac{1}{3}}

: cube it and you get

8^1 = 8

, so

8^{\frac{1}{3}} = \sqrt[3]{8} = 2

n-th root

\sqrt[n]{a}

is the number which, raised to the power n, gives a. For example

\sqrt[3]{64} = 4

because

4^3 = 64

Rational exponent

an exponent that is a fraction

\dfrac{m}{n}

(m and n whole numbers). The bottom number n is the root, the top number m is the power.

Surd

a root that is not a whole number, such as

\sqrt{2}

. With the new notation,

\sqrt{2} = 2^{\frac{1}{2}}

★ Rational exponents

$a^{\frac{1}{n}} = \sqrt[n]{a}$ (for example $x^{\frac{1}{2}} = \sqrt{x}$ and $x^{\frac{1}{3}} = \sqrt[3]{x}$ )
$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$ (root and power can be done in either order; root first keeps numbers small)
$a^{-\frac{m}{n}} = \dfrac{1}{a^{\frac{m}{n}}}$ (a negative exponent means flip, it does not make the answer negative)
All the laws still work: $a^p \times a^q = a^{p+q}$ , $\dfrac{a^p}{a^q} = a^{p-q}$ , $(a^p)^q = a^{pq}$ , $(ab)^p = a^p b^p$
In the IB course the base is positive, so a^{$\dfrac{m}{n}$} is always positive too.

⚠️ Watch out

$a^{\frac{1}{2}}$ means $\sqrt{a}$ . It does not mean $\dfrac{a}{2}$ , and $a^{\frac{1}{3}}$ is not $\dfrac{a}{3}$ .
$8^{-\frac{2}{3}}$ is not negative. It equals $\dfrac{1}{8^{\frac{2}{3}}} = \dfrac{1}{4}$ .
You cannot combine terms with different powers: $x^{\frac{1}{2}} + x^{\frac{1}{3}}$ does not simplify. The laws work for multiplying and dividing (and powers), never for adding.
$\sqrt{x^2 + y^2}$ is not $x + y$ . A root does not distribute over addition.

💡 Command term: "simplify" / "write in the form kx^n"

Turn every root into a power, collect numbers and powers of x separately, then use the laws. Give the final answer as one number times one power of x, such as

5x^{\frac{5}{2}}

, with fractional exponents left as fractions.

🧪 Try it yourselfHow does y = x^p change with p?

Drag p. At p = 0.5 you see the square-root curve, at p = 1 a straight line and at p = 2 a parabola. Every curve passes through (1, 1) because 1 to any power is 1. Try p = 0.33 and p = 0.75 as well. These are rational exponents.

Worked examples

Example 1.1Roots written as powers

(a) Evaluate

49^{\frac{1}{2}}

125^{\frac{1}{3}}

16^{\frac{1}{4}}

and

\left(\dfrac{1}{32}\right)^{\frac{1}{5}}

.
(b) Write

\sqrt{x}

\sqrt[3]{x^2}

and

\dfrac{1}{\sqrt{x}}

as powers of x.

🧠 Think it through

(a) The exponent $\dfrac{1}{n}$ means "the n-th root". Ask: which number, multiplied by itself n times, gives the base?
(b) Use $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ . A root on the bottom of a fraction gives a negative exponent.

✍️ By hand
(a)    (since )
   (since )
   (since )
   (since )
(b)    and   

🔢 On your TI-Nspire CX II

Type the power with the fraction in brackets.

✅ Answer (a) 7, 5, 2 and

\dfrac{1}{2}

. (b)

x^{\frac{1}{2}}

x^{\frac{2}{3}}

x^{-\frac{1}{2}}

Brackets matter! 49^

\dfrac{1}{2}

on the calculator means

\dfrac{49^1}{2} = 24.5

, which is wrong. Always write

49^{\frac{1}{2}}

Example 1.2Evaluating

\dfrac{m}{n}

powers and negative fractional powers

Evaluate without a calculator: (a)

8^{\frac{2}{3}}

(b)

16^{\frac{3}{4}}

(c)

125^{-\frac{2}{3}}

(d)

\left(\dfrac{9}{4}\right)^{-\frac{3}{2}}

🧠 Think it through

Split the exponent: $a^{\frac{m}{n}}$ : take the n-th root first, then raise to the power m.
For a negative exponent, flip first (reciprocal), then deal with the positive exponent.
For a fraction base, apply the root to the top and the bottom.

✍️ By hand
(a) 
(b) 
(c) 
(d) 

🔢 On your TI-Nspire CX II

Check on a Calculator page. Use brackets around every fraction and a negative-key (−)💻 type - for the minus sign. Press ctrl💻 Ctrl enter ↵💻 Enter if you want a decimal instead of a fraction.

✅ Answer (a) 4 (b) 8 (c)

\dfrac{1}{25}

(d)

\dfrac{8}{27}

On the screen (c) is 0.04 =

\dfrac{1}{25}

and (d) is 0.296296 =

\dfrac{8}{27}

. In an exam, "write down the exact value" wants the fraction.

Example 1.3Simplify and write in the form kx^n

Write each expression in the form

kx^n

:
(a)

5x^2\sqrt{x}

(b)

\left(8x^6\right)^{\frac{1}{3}}

(c)

\dfrac{6x}{\sqrt[3]{x}}

(d)

\dfrac{x^2\sqrt[3]{x}}{\sqrt{x}}

🧠 Think it through

Change every root into a power of x.
Multiply: add the exponents. Divide: subtract the exponents. Power of a power: multiply the exponents.
Work out fractions of exponents with a common denominator. (d): 2 + $\dfrac{1}{3}$ − $\dfrac{1}{2}$ .

✍️ By hand
(a) 
(b) 
(c) 
(d) 

🔢 On your TI-Nspire CX II

Check any one of them numerically by trying x = 4 in the original and in your answer. For (c):

✅ Answer (a)

5x^{\frac{5}{2}}

(b) 2x² (c)

6x^{\frac{2}{3}}

(d)

x^{\frac{11}{6}}

Choosing a test value like x = 4 is a quick way to catch an arithmetic slip. If both lines give the same number, your simplification is almost certainly right.

Example 1.4Solving equations with rational exponents

Solve for x (x > 0): (a)

x^{\frac{3}{2}} = 27

(b)

2x^{-\frac{1}{2}} = 0.5

(c)

x^{\frac{2}{3}} = 25

(d)

(x - 1)^{\frac{1}{3}} = 2

🧠 Think it through

Isolate the power first (divide by 2 in part b).
To cancel the exponent $\dfrac{m}{n}$ , raise both sides to the reciprocal $\dfrac{n}{m}$ . Reason: $(x^{\frac{m}{n}})^{\frac{n}{m}} = x^1$ .
In (d) the base is the bracket (x − 1). Cube both sides.

✍️ By hand
(a) 
(b) 
(c) 
(d) 

🔢 On your TI-Nspire CX II

Check by substituting your answers back in.

✅ Answer (a) x = 9 (b) x = 16 (c) x = 125 (d) x = 9

Check: 9 is positive and

9^{\frac{3}{2}}

= 27 ✓. Always substitute back, because raising to a power can sometimes bring in extra answers.

Example 1.5Metabolic rate and body mass

A model for the daily energy use R (kcal per day) of a mammal with mass m kg is

R = 70\,m^{\frac{3}{4}}

(a) Find R for a person of mass 70 kg and for an elephant of mass 4000 kg.
(b) Find the mass of an animal that uses 1400 kcal per day.
(c) By what percentage does R increase when the mass doubles?

🧠 Think it through

(a) Substitute. The exponent $\dfrac{3}{4}$ = 0.75, so type it in brackets.
(b) Solve $70m^{\frac{3}{4}} = 1400$ : divide by 70, then raise both sides to the power $\dfrac{4}{3}$ .
(c) Compare R(2m) with R(m): $70(2m)^{\frac{3}{4}} = 2^{\frac{3}{4}} \times 70m^{\frac{3}{4}}$ .

✍️ By hand
(a) 
 kcal per day (person)
 kcal per day (elephant)
(b) 
 kg
(c) , so R is multiplied by 1.682: an increase of 68.2%.

🔢 On your TI-Nspire CX II

Type each calculation with the fractional exponent in brackets.

✅ Answer (a) 1690 kcal per day (3 s.f.) and 35 200 kcal per day (b) 54.3 kg (c) 68.2%

Notice that the elephant is 57 times heavier but needs only about 21 times the energy. Because

\dfrac{3}{4}

is less than 1, big animals are more energy-efficient per kilogram.

Challenge questions

⭐ Challenge 1.6A quotient with rootsPaper 1 style

Write

\dfrac{\left(16x^8\right)^{\frac{3}{4}}}{4\sqrt{x}}

in the form

kx^n

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Top: $(16x^8)^{\frac{3}{4}} = 16^{\frac{3}{4}}\,x^{8 \times \frac{3}{4}}$ .
Bottom: $4\sqrt{x} = 4x^{\frac{1}{2}}$ .
Divide the numbers, subtract the exponents.

✍️ By hand

16^{\frac{3}{4}} = 8

and

x^{8 \times \frac{3}{4}} = x^6

\dfrac{8x^6}{4x^{\frac{1}{2}}}

\quad = 2x^{6 - \frac{1}{2}}

\quad = 2x^{\frac{11}{2}}

🔢 On your TI-Nspire CX II

Check with x = 4: both lines should give the same number.

✅ Answer

2x^{\frac{11}{2}}

(so k = 2, n =

\dfrac{11}{2}

= 5.5)

⭐ Challenge 1.7Species on an islandPaper 2 style

Ecologists model the number of species S on an island of area A km² by

S = 4A^{\frac{1}{4}}

(a) Find S for an island of area 81 km² and for one of area 625 km².
(b) An island has 20 species. Find its area.
(c) Find the area of an island expected to have 10 species.
(d) Find the percentage increase in S when the area doubles.
(e) Show that the area must be multiplied by 16 to double the number of species.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

(a) Substitute. $81^{\frac{1}{4}} = 3$ .
(b),(c) Divide by 4, then raise both sides to the power 4.
(d) Compare $S(2A)$ with $S(A)$ : $4(2A)^{\frac{1}{4}} = 2^{\frac{1}{4}} \times 4A^{\frac{1}{4}}$ .
(e) Want $S_{new} = 2S$ . Try area kA: $k^{\frac{1}{4}} = 2$ .

✍️ By hand
(a)  and 
(b) 
 km²
(c) 
 km²
(d) , so S rises by 18.9%
(e) 

🔢 On your TI-Nspire CX II

Use the calculator for (a), (c) and (d).

✅ Answer (a) 12 species and 20 species (b) 625 km² (c) 39.1 km² (d) 18.9% (e)

k^{\frac{1}{4}}

= 2 gives k = 16, shown.

Next: Lesson 2 · Laws of logarithms

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL HL · Logs & series 1 · Rational exponents: roots as powers

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

Rational exponents: roots as powers

Worked examples

Challenge questions