Exponent Laws
Match the Exponent Law
Draw a line from each law to an example of it.
Multiply with Same Base
Simplify using aᵐ × aⁿ = aᵐ⁺ⁿ.
3² × 3⁴
5³ × 5²
x⁵ × x³
2⁴ × 2
Divide with Same Base
Simplify using aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
4⁶ ÷ 4²
x⁷ ÷ x³
3⁵ ÷ 3⁵
y⁹ ÷ y²
Power of a Power
Simplify using (aᵐ)ⁿ = aᵐⁿ.
(2³)²
(x²)⁵
(3²)³
(a⁴)⁴
Which Law?
Sort each expression by which exponent law applies.
Match the Simplification
Draw a line to match each expression to its simplified form.
Exponential Growth Pattern
Each sequence follows a constant-multiply pattern. Write the next two values.
Mixed Exponent Practice
Apply the correct law to simplify each expression.
2³ × 2² ÷ 2⁴
(3²)³ ÷ 3⁴
x⁶ × x² ÷ x³
(a²)³ × a⁰
Zero and One Exponents
Evaluate each expression.
7⁰ =
(−3)⁰ =
x⁰ (for any x ≠ 0) =
8¹ =
(abc)⁰ =
Negative Exponents
Draw a line to match each negative exponent expression to its equivalent fraction.
Negative Exponent Introduction
Use a⁻ⁿ = 1/aⁿ. Circle the correct equivalent.
2⁻³ =
5⁻² =
10⁻² =
3⁻¹ =
Positive, Negative or Zero Exponent?
Sort each expression by its exponent type.
Scientific Notation Connection
Draw a line to match each number to its scientific notation.
Standard Form or Not?
Sort each number: is it correctly written in scientific notation (a × 10ⁿ where 1 ≤ a < 10)?
Convert from Scientific Notation
Write each number in standard (ordinary) form.
3.2 × 10³ =
6.5 × 10⁻² =
1.04 × 10⁵ =
2.7 × 10⁻⁴ =
Multiply in Scientific Notation
Multiply and give the answer in correct scientific notation.
(2 × 10³) × (3 × 10²)
(5 × 10⁴) × (2 × 10³)
(3 × 10⁻²) × (4 × 10⁵)
(2.5 × 10²) × (4 × 10³)
Explain and Apply
Show all working.
Simplify: (2x³)² × x⁴. Show each step and name each law used.
A bacteria culture doubles every hour. After t hours there are 2ᵗ bacteria. How many after 8 hours? Express as a power of 2 and calculate the actual number.
Simplify: (3a²b)³. Show full working.
Word Problems with Exponents
Show all working and express answers as powers where possible.
A town's population doubles every 10 years. It currently has 5 000 people. Write an expression for the population after 30 years and calculate it.
Simplify: (3² × 3³) ÷ 3⁴. Show each step and name the law used.
Order These Expressions from Smallest to Largest
Evaluate each expression, then sort from smallest to largest.
Fractional Exponents (Introduction)
A fractional exponent means a root. a^(1/2) = √a, a^(1/3) = ∛a.
9^(1/2) =
8^(1/3) =
25^(1/2) =
27^(1/3) =
Exponents in Context — Science
Apply exponent skills to real science contexts.
The mass of a proton is approximately 1.67 × 10⁻²⁷ kg. The mass of an elephant is approximately 5 × 10³ kg. How many times heavier is the elephant than the proton? Express your answer in scientific notation.
The distance from the Earth to the Sun is approximately 1.5 × 10¹¹ metres. Light travels at 3 × 10⁸ m/s. How long does it take light to travel from the Sun to the Earth? Express in seconds and in minutes.
Expressions in Ascending Order
Evaluate each expression, then sort from smallest to largest value.
Applying All Exponent Laws Together
Simplify each expression using the correct combination of laws.
(x³)² × x⁻² =
a⁵ × a⁻³ ÷ a =
(2b²)³ ÷ 4b =
m⁰ × m³ × m⁻² =
Write and Simplify
Write each situation as an expression using exponents, then simplify.
A square has side x³ cm. Write an expression for its area. Simplify.
A cube has side 2a² m. Write an expression for its volume. Simplify.
Simplify: (4x²y)² ÷ (2xy)². Show all steps.
Divide in Scientific Notation
Divide and give the answer in correct scientific notation.
(9 × 10⁶) ÷ (3 × 10²)
(8 × 10⁵) ÷ (4 × 10⁷)
(6 × 10⁻³) ÷ (2 × 10²)
(1.5 × 10⁸) ÷ (5 × 10³)
Scientific Notation Word Problems
Use scientific notation throughout. Show all working.
The human body contains about 3.7 × 10¹³ cells. A typical cell is about 1 × 10⁻⁵ m across. If all cells were laid end to end, what would the total length be? Give your answer in metres using scientific notation.
Australia's GDP is approximately $2.2 × 10¹² dollars and its population is approximately 2.6 × 10⁷ people. Calculate the GDP per person in dollars. Round to 3 significant figures.
Add and Subtract in Scientific Notation
Convert so exponents match, then add or subtract.
3 × 10⁴ + 2 × 10³ =
7 × 10⁵ − 4 × 10⁵ =
6 × 10⁶ + 5 × 10⁵ =
9 × 10³ − 3 × 10² =
Exponential Equations (Introduction)
Use laws to find the value of the unknown exponent.
Find x: 2ˣ = 32. (Hint: write 32 as a power of 2.)
Find x: 3ˣ = 81.
Find x: 10ˣ = 0.001.
Find x: 5ˣ = 1/125.
Match Equation to Solution Method
Sort each equation type by how you would solve it.
Proof and Justification
Prove each result using exponent laws and numerical examples.
Prove that (ab)ⁿ = aⁿbⁿ using the definition of exponents. (Hint: write out (ab)ⁿ as repeated multiplication, then rearrange.)
Show that a⁻ⁿ = 1/aⁿ using the quotient law. Start with aⁿ ÷ aⁿ = a⁰ = 1, then divide both sides by aⁿ.
Compound Interest Application
Apply the formula A = P(1 + r)ⁿ using a calculator.
You invest $2000 at 6% per year compounded annually for 5 years. Calculate A = 2000 × (1.06)⁵. Round to the nearest cent. How much interest did you earn?
Which is better: 8% simple interest on $500 for 4 years, or 7% compound interest on $500 for 4 years? Show all calculations.
Simplifying Complex Expressions
Simplify each expression completely. Show every step and name each law.
Simplify: (3a²)³ × (2a)⁻² ÷ a⁵
Simplify: [(x²y³)²]³ × (x⁻¹y)⁴. Write the answer with positive exponents.
Exponential Modelling
Write and solve an exponential model for each scenario.
A radioactive substance decays by half every year. Starting with 80 g, write an expression for the amount remaining after n years. How much remains after 5 years?
A city's population grows at 3% per year. Starting at 200 000, write an expression for the population after n years. Predict the population after 10 years. Round to the nearest thousand.
Pitfall Prevention — Which is Correct?
Identify the correct simplification.
3² × 4² =
3² + 4² =
(3 × 4)² =
3⁵ + 3⁵ =
Generalising — Create Your Own Problems
Write your own example of each law, then swap with a family member to solve.
Write one example each of the product law, quotient law, and power law using the variable m. Solve each one yourself first.
Write one multi-step problem that uses at least two laws. Include negative exponents or scientific notation. Provide a full solution.
Investigation: Towers of Powers
Explore patterns in towers of exponents.
Evaluate: 2^(2^1), 2^(2^2), 2^(2^3). Write each result. Is the pattern exponential? How fast does it grow?
The expression 2^(3^2) could mean (2³)² = 64 or 2^(3²) = 2⁹ = 512. These are different! Use exponent laws to explain which one is larger. Which convention do mathematicians use?
Error Spotting
Find the error and write the correct answer.
Student: 'x² × x³ = x⁶ because 2 × 3 = 6.' What is wrong? What is the correct answer?
Student: '(x³)² = x⁵ because 3 + 2 = 5.' What is wrong? What is the correct answer?
Student: 'x⁵ ÷ x⁵ = 0 because 5 − 5 = 0.' What is wrong? What is the correct answer?
Scientific Notation Conversions
Convert between standard and scientific notation.
Write in scientific notation: 0.000045, 678 000 000, 0.00000001.
Write in standard form: 3.7 × 10⁴, 8.2 × 10⁻³, 1.05 × 10⁶.
Negative Exponents — Problem Set
Evaluate each expression. Express answers as fractions, then decimals.
Evaluate: 5⁻², 3⁻³, 10⁻⁴. Express as fractions and decimals.
Simplify: x⁻² × x⁵, y³ × y⁻⁷, (a²)⁻³.
Exponents in Real Life
Explore exponential patterns in everyday situations.
- 1Look up the current savings account interest rate at an Australian bank. Use A = P(1 + r)ⁿ to calculate how much $1000 would grow to after 5, 10, and 20 years. Plot these three points on a number line.
- 2Research Avogadro's number (6.022 × 10²³). Find out what it represents, then calculate how many molecules are in 18 g of water. Express your answer in scientific notation.
- 3Find a real-world example of exponential growth or decay (population, radioactive decay, viral spread). Write the exponential expression, identify the base and exponent, and describe what they represent.