Exponents & logs 1 · Powers and the three basic laws

Module 5 of 47Lesson 1 of 7 · Exponents & logs 1 · Powers and the three basic laws

Exponents & logarithms · Essential understanding 1

Powers and the three basic laws

IB AI SL 1.5with 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: An exponent (or power) is a short way to write repeated multiplication. Three simple laws let you multiply, divide and "power up" expressions without writing everything out.

By the end of this lesson you can:

explain what the base and the exponent mean in aⁿ
use the product, quotient and power laws with numbers and letters
use the zero exponent: a⁰ = 1
use the laws "backwards" and rewrite different bases as the same base
check an answer on the TI-Nspire CX II

🧑🏾 Imagine this…

Neo sends a voice note to 3 friends. Each of those friends forwards it to 3 new people. Then each of them forwards it to 3 more. After 4 rounds, how many people got it in the last round?
You could write 3 × 3 × 3 × 3. That is tiring! Mathematicians write it as 3⁴. The little 4 says "multiply four 3s together".

Base

the number (or letter) that is being multiplied. In 3⁴ the base is 3.

Exponent

the little raised number. It tells you HOW MANY of the base to multiply. In 3⁴ the exponent is 4. (Also called power or index.)

Power

the whole thing: 3⁴ is "3 to the power 4". Its value is 3 × 3 × 3 × 3 = 81.

Coefficient

the number in front of a letter. In 7b³ the coefficient is 7.

★ The three basic laws (same base!)

Product law: $a^m \times a^n = a^{m+n}$ (multiplying: add the exponents)
Quotient law: $\dfrac{a^m}{a^n} = a^{m-n}$ (dividing: subtract the exponents)
Power law: $(a^m)^n = a^{mn}$ (power of a power: multiply the exponents)
Zero exponent: $a^0 = 1$ for any $a \neq 0$ .
A letter on its own has exponent 1: x = x¹.

Why do they work? $2^3 \times 2^4 = (2\times2\times2)\times(2\times2\times2\times2) = 2^7$ : three 2s and four 2s make seven 2s, so we just add 3 + 4. For $\dfrac{2^5}{2^2}$ two of the five 2s cancel with the bottom, leaving three 2s: 5 − 2 = 3. And $(2^3)^2 = 2^3 \times 2^3 = 2^6$ : two lots of three 2s, so 3 × 2 = 6. Finally, setting m = n in the quotient law gives $\dfrac{a^n}{a^n} = a^0$ , but anything divided by itself is 1, so $a^0 = 1$ .

⚠️ Watch out

The laws only work when the bases are the same. 2³ × 3⁴ can NOT be turned into one power just by adding.
Add exponents when you multiply powers: $a^3 \times a^4 = a^7$ , NOT $a^{12}$ . Multiply exponents only for a power of a power: $(a^3)^4 = a^{12}$ .
Multiply or divide the numbers in front separately: 3a⁴ × 5a⁶ = 15a¹⁰ (the 15 comes from 3 × 5, NOT from adding).

💡 Command term: "simplify"

"Simplify" means write the answer as one neat power of each letter, with the number in front. Do not multiply out big numbers unless asked to "evaluate" or "give the value".

🧪 Try it yourselfSpot the correct law

Pick the correct simplification. Think: add, subtract or multiply the exponents?

Worked examples

Example 1.1The three basic laws

Simplify, leaving your answers as a single power. Do not evaluate.
(a)

m^5 \times m^4

(b)

\dfrac{p^{11}}{p^4}

(c)

(5^2)^6

🧠 Think it through

Check the bases are the same: m and m; p and p; 5 and 5. Yes.
(a) multiplying → add the exponents.
(b) dividing → subtract the exponents.
(c) power of a power → multiply the exponents.

✍️ By hand
(a) 
(b) 
(c) 

🔢 On your TI-Nspire CX II

Check a law with numbers. On a Calculator page type 2^5*2^4 enter ↵💻 Enter, then 2^9 enter ↵💻 Enter. Both give the same value, so $2^5 \times 2^4 = 2^{9}$ .
Use the ^ key for powers.

✅ Answer (a) m⁹ (b) p⁷ (c) 5¹²

Checking a law with small numbers on your GDC is a quick way to catch a slip when you are unsure.

Example 1.2Numbers in front of the letters

Simplify: (a)

3a^4 \times 5a^6

(b)

\dfrac{12b^9}{4b^3}

(c)

4d^2 \times 3d \times d^5

🧠 Think it through

Deal with the numbers on their own, then deal with each letter on its own.
(a) numbers: 3 × 5 = 15. Letters: a⁴ × a⁶.
(b) numbers: $\dfrac{12}{4} = 3$ . Letters: $\dfrac{b^{9}}{b^{3}}$ .
(c) A lone d means d¹. Numbers: 4 × 3 = 12 (and × 1).

✍️ By hand
(a) 
(b) 
(c) 

🔢 On your TI-Nspire CX II

Check (a) by putting a = 2. The two lines must match.

✅ Answer (a) 15a¹⁰ (b) 3b⁶ (c) 12d⁸

Example 1.3The zero exponent

Write down the value of (a)

9^0

(b)

(5x)^0

(c)

5x^0

(d)

\dfrac{7^6}{7^6}

🧠 Think it through

Anything (except 0) to the power 0 equals 1.
In (b) the exponent 0 covers the whole bracket (5x). In (c) it covers only the x.
(d) is a number divided by itself, so it must be 1. The quotient law gives the same answer.

✍️ By hand
(a) 
(b) 
(c) 
(d) 

🔢 On your TI-Nspire CX II

Type 9^0 and 5*3^0 to see the idea (using x = 3).

✅ Answer (a) 1 (b) 1 (c) 5 (d) 1

Example 1.4Using the laws backwards: same base

(a) Write

2^{12}

as a power of 8.
(b) Write

8^3 \times 4^5

as a single power of 2.
(c) Simplify

\dfrac{125^2}{25^3}

🧠 Think it through

You must make the bases the same first. Look for the small prime: 8 = 2³, 4 = 2², 25 = 5², 125 = 5³.
(a) 2¹² = 2^3×4, so it is (2³)⁴.
(b) Rewrite 8 and 4 as powers of 2, use the power law, then the product law.
(c) Rewrite both as powers of 5, then divide.

✍️ By hand
(a) 
(b) 
(c) 

🔢 On your TI-Nspire CX II

Check each result with numbers.

✅ Answer (a) 8⁴ (b) 2¹⁹ (c) 1

In (c) the answer 1 is not a mistake. Big numbers can cancel exactly.

Example 1.5Forwarding messages: a growth context

Thato posts a message. In round 1 it reaches 3 people. In each round, every person who got it in the previous round shares it with 3 new people. So in round n,

3^n

people receive it.
(a) How many people receive it in round 4?
(b) In round 2, 3² people receive it. Over the next 3 rounds the number is multiplied by 3³. Write the round 5 total as a product of two powers of 3 and simplify.
(c) Find

\dfrac{3^7}{3^4}

and explain what it means.

🧠 Think it through

(a) Put n = 4 into 3ⁿ.
(b) Round 2 has 3² people. Three rounds later the number has been multiplied by 3³. So multiply: 3² × 3³.
(c) Use the quotient law. It compares round 7 with round 4.

✍️ By hand
(a) 
(b) 
 people in round 5
(c) 

🔢 On your TI-Nspire CX II

Type the values to check.

✅ Answer (a) 81 people. (b) 3⁵ = 243 people. (c) 27: round 7 has 27 times as many receivers as round 4. Each 3 rounds multiplies the number by 3³ = 27.

Notice the pattern: adding to the exponent means multiplying the number of people. This idea becomes exponential growth in Lesson 4.

Challenge questions

⭐ Challenge 1.6Simplify with numbers in frontPaper 1 style

Simplify

\dfrac{6x^7 \times 3x^2}{9x^4}

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

Show the full solution

🧠 Think it through

Top first: numbers 6 × 3 = 18; letters x⁷ × x² = x⁹.
Now divide by 9x⁴: numbers $\dfrac{18}{9} = 2$ ; letters $\dfrac{x^{9}}{x^{4}} = x^{5}$ .

✍️ By hand

\dfrac{6x^7 \times 3x^2}{9x^4} = \dfrac{18x^9}{9x^4}

\quad = 2x^5

🔢 On your TI-Nspire CX II

Check with x = 2.

✅ Answer 2x⁵

⭐ Challenge 1.7A rectangular tankPaper 2 style

A tank has length

4x^3

cm, width

5x^2

cm and height

2x

cm, where x > 0.

(a) Find and simplify an expression for the area of the base.
(b) Find and simplify an expression for the volume V.
(c) Find and simplify $\dfrac{V}{\text{base area}}$ . What does this tell you?
(d) Find the volume when x = 3.
(e) The volume is 2560 cm³. Find x.

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

Show the full solution

🧠 Think it through

Base area = length × width.
Volume = base area × height. Multiply numbers, add exponents.
Divide the two expressions. The answer should be the height!
Substitute x = 3 into your volume expression.
Solve 40x⁶ = 2560: divide, then use the GDC solver (or think: which x makes x⁶ = 64?).

✍️ By hand
(a) 
(b) 
(c) , the height of the tank ().
(d) 
(e) 

🔢 On your TI-Nspire CX II

Go to menu💻 click menu on keypad▸3: Algebra▸1: Numerical Solve.
Type nSolve(40*x^6=2560,x=1) enter ↵💻 Enter. The part x=1 is a starting guess; choose a positive one because x > 0.
📍 Where to find it on the TI-Nspire
nSolve( is in menu💻 click menu on keypad▸3: Algebra▸1: Numerical Solve

✅ Answer (a) 20x⁵ cm² (b) 40x⁶ cm³ (c) 2x cm: the height (d) 29 160 cm³ (e) x = 2

Always give units: cm² for area, cm³ for volume.

Next: Lesson 2 · Negative powers, brackets, fractions & standard form

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL Exponents & logs 1 · Powers and the three basic laws

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

Powers and the three basic laws

Worked examples

Challenge questions