Number

Exponent Laws

Name That Law

Draw a line from each example to the exponent law it demonstrates.

a^3 x a^4 = a^7

a^6 / a^2 = a^4

(a^3)^2 = a^6

a^0 = 1

a^-2 = 1/a^2

Product law

Quotient law

Power of a power law

Zero index law

Negative index law

Apply the Product Law

Circle the correct simplified form for each expression (a^m x a^n = a^(m+n)).

x^3 x x^5

x^8

x^15

x^2

2^3 x 2^4

2^12

2^7

4^7

y^2 x y x y^3

y^6

y^5

y^7

3^2 x 3^3

9^5

3^6

3^5

Apply the Quotient Law

Circle the correct simplified form (a^m / a^n = a^(m-n)).

x^7 / x^3

x^4

x^10

x^21

2^5 / 2^2

2^3

2^10

1^3

a^8 / a^8

a^0 = 1

a^64

a^16

Simplify Using Index Laws

Simplify each expression. Show the law you are using.

m^5 x m^3 =

n^9 / n^4 =

(p^2)^4 =

3x^2 x 4x^3 =

(2a^3)^2 =

Negative and Zero Indices

Evaluate or simplify. Write fractions where necessary.

5^0 =

3^-2 =

2^-3 =

x^-1 written as a fraction:

Group by Index Law Used

Sort each expression into the correct column according to the law needed.

a^4 x a^2

(b^3)^5

c^6 / c^2

d^2 x d^7

(e^4)^3

f^8 / f^3

Product Law

Quotient Law

Power of Power

Mixed Index Law Challenge

Simplify each expression fully. Show all steps.

(3x^2 y)(2x^3 y^4) =

12a^6 b^4 / (4a^2 b) =

(2m^3)^3 / m^5 =

Index Laws in the Real World

Try these index law investigations at home.

1Research how computer storage (bytes, kilobytes, megabytes, gigabytes) uses powers of 2. Write each unit as an index expression.
2Create your own set of 3 simplify-me questions using the product law, then swap with a family member to solve.
3Write the current year as a product of prime factors using index notation.

Index Laws with Variables -- Product and Quotient

Apply the product and quotient laws to these variable expressions. Show the law you use.

x^3 x x^5 =

y^8 / y^3 =

a^4 x a^-2 =

b^5 / b^-1 =

Negative Exponents with Variables

Write each expression without negative exponents (as a fraction).

x^-3 =

3y^-2 =

a^-1 b^2 =

(2x)^-2 =

Fractional Bases

Evaluate each expression. Write as a fraction or decimal.

(1/2)^3 =

(3/4)^2 =

(2/3)^3 =

(1/5)^2 =

Comparing Exponential Expressions

Without a calculator, circle the greater expression in each pair.

2^10 or 10^2?

2^10 is greater

10^2 is greater

They are equal

3^-1 or 3^0?

3^-1 is greater

3^0 is greater

They are equal

(1/2)^2 or (1/3)^2?

(1/2)^2 is greater

(1/3)^2 is greater

They are equal

5^2 or 2^5?

5^2 is greater

2^5 is greater

They are equal

Continuing Exponential Sequences

Find the next two terms in each sequence. Use index laws to explain the pattern.

125

Product Law — Extended Practice

Apply the product law a^m x a^n = a^(m+n). Simplify each expression.

x^4 x x^6 =

5^3 x 5^4 =

a^2 x a^3 x a^5 =

2^2 x 2^3 x 2 =

3x^4 x 5x^2 = (multiply coefficients separately)

4a^3 b^2 x 3a b^4 =

TipThe product law only applies when the BASES are the same. 2^3 x 3^2 cannot be simplified using the product law because the bases (2 and 3) differ.

Quotient Law — Extended Practice

Apply the quotient law a^m / a^n = a^(m-n). Simplify each expression.

x^9 / x^3 =

7^8 / 7^5 =

a^4 / a^6 = (hint: result will have a negative exponent)

12x^7 / 4x^2 =

a^3 b^5 / (a^2 b^3) =

TipIf the result of subtracting exponents gives a negative number, the answer should be written as a negative index or a fraction. Both forms are correct.

Power of a Power — Extended Practice

Apply (a^m)^n = a^(mn). Simplify each expression.

(x^3)^4 =

(2^3)^4 =

(a^5)^2 =

(x^-2)^3 =

((m^3)^2)^2 =

Power of a Product Rule

Use (ab)^n = a^n b^n to expand each expression.

(2x)^3 =

(3y)^4 =

(xy)^5 =

(2ab)^2 =

(-3x^2)^3 =

TipThis extends the power of a power law. When an entire product is raised to a power, each factor gets that power. For example: (2x)^3 = 2^3 x x^3 = 8x^3.

Evaluating Negative Indices Numerically

Evaluate each expression as a fraction or decimal. No variables — just numbers.

2^-4 =

5^-2 =

3^-3 =

10^-5 =

(1/3)^-2 = (Hint: reciprocal rule)

TipRemind your child: a^-n = 1/a^n. So 4^-2 = 1/4^2 = 1/16. Once the rule is clear, these become straightforward fraction calculations.

Mixed Index Law — Choose the Correct Law

Circle the correct name of the index law being used.

Simplify (p^4)^3 = p^12

Product law

Power of power law

Quotient law

Simplify x^5 / x^3 = x^2

Quotient law

Product law

Negative index law

Simplify 7^0 = 1

Zero index law

Negative index law

Product law

Simplify m^-3 = 1/m^3

Negative index law

Zero index law

Quotient law

Multi-Law Simplification

Each expression requires more than one index law. Identify the steps and simplify fully.

(x^3)^2 x x^4 = (Step 1: power of power) = ___ x x^4 = (Step 2: product law) =

(2a^2)^3 / (4a^4) =

(x^4 y^2)^3 / (x^3 y) =

(3m^2 n)^2 x (2mn^3) =

Exponential Growth Calculations

Use index laws to calculate the results of exponential growth.

A population of bacteria doubles every hour. Starting with 1 bacterium, after 12 hours how many bacteria are there? Write as a power of 2 and in standard form.

A computer virus replicates: each copy creates 3 new copies every minute. Starting with 1 copy, after 5 minutes how many copies are there? Write as a power of 3.

If the bacteria population from the first question doubles again for another 6 hours, how many are there at the end of 18 hours total? Write using index laws: 2^12 x 2^6 =

TipThese problems apply index laws to real-world exponential models. The link between abstract algebra and practical applications helps students see the relevance of the skills they are practising.

Expression to Simplified Form

Draw a line from each expression to its simplified form.

a^7 x a^-3

(b^3)^4

c^0 + c^0

d^-2 x d^5

(2e^3)^2

a^4

b^12

d^3

4e^6

TipThis self-checking matching activity is great preparation for exam conditions. If a match does not appear, your child has made an error — encourage them to recheck before asking for help.

Index Laws with Multiple Variables

Simplify each expression. Multiple variables each obey index laws independently.

x^3 y^4 x x^2 y =

(a^2 b^3)(a^3 b^2) =

x^6 y^4 / (x^2 y^2) =

(m^2 n^3)^4 =

(3x^2 y^3)(2xy^2) / (6x^2 y^4) =

Fraction Bases and Index Laws

Evaluate each expression with a fractional base.

(2/3)^3 =

(3/4)^-2 = (Hint: flip the fraction, then square)

(1/2)^-4 =

(5/2)^2 / (5/2)^3 =

TipThe index laws apply to all bases, including fractions. For (a/b)^n = a^n / b^n. For (a/b)^-n = (b/a)^n — the reciprocal is raised to the positive power.

Simplify Expressions Involving Negative Indices

Simplify each expression, writing the answer with only positive indices.

x^-3 y^4 / (x^2 y^-2) =

(a^-2 b^3)^2 =

m^5 n^-3 x m^-2 n^4 =

(2x^-1)^3 / x^2 =

Proving Index Law Generalisations

Prove each law by expanding and simplifying a specific numerical example.

Prove a^m x a^n = a^(m+n) using a = 2, m = 3, n = 4: LHS = ___ x ___ = ___. RHS = 2^___ = ___. Equal? ___

Prove (a^m)^n = a^(mn) using a = 3, m = 2, n = 3: LHS = (___)^___ = ___. RHS = 3^___ = ___. Equal? ___

Prove a^0 = 1 using a = 5: 5^2 / 5^2 = 5^(___ - ___) = 5^0. But 5^2 / 5^2 = ___. Therefore 5^0 = ___.

TipChecking a law with specific numbers builds confidence that the abstract rule is correct. This habit — checking a general rule with a specific example — is useful throughout mathematics.

Which Is Larger?

Circle the larger expression in each pair.

3^4 or 4^3

3^4 (= 81)

4^3 (= 64)

They are equal

2^10 or 10^3

2^10 (= 1024)

10^3 (= 1000)

They are equal

5^-1 or (1/5)^-1

5^-1 = 1/5

(1/5)^-1 = 5

They are equal

(-2)^4 or (-2)^3

(-2)^4 = 16

(-2)^3 = -8

They are equal

Index Laws Applied to Scientific Notation

Use index laws to perform these scientific notation calculations.

(10^3)^4 = 10^___ = (write in standard form)

(2 x 10^3)^3 = 2^___ x 10^___ = ___ x 10^___

(4 x 10^6) x (3 x 10^-2) — use product law for the powers of 10: =

(9 x 10^8)^(1/2) — what would this mean? (Hint: (a^8)^(1/2) = a^4.) =

TipThis activity connects index laws to the preceding scientific notation topic — a great opportunity to discuss how these topics relate to each other.

Solving Index Equations

Solve for x in each equation. Use index laws to find x.

2^x = 16. Write 16 as a power of 2: x =

3^x = 27. x =

5^x = 1/25 = 5^___. x =

x^3 = 64. x =

2^x x 2^3 = 2^8. x =

TipSolving index equations extends index law knowledge to algebraic thinking. If your child struggles, suggest they try different values of x and use the equation to check.

Sort Expressions by Value (Ascending)

Sort these expressions from smallest value to largest. Evaluate each first.

2^0

2^-2

2^3

2^-1

2^2

Smallest

2nd

3rd

4th

Largest

Applying Index Laws — Real Context

Answer each problem using index laws. Show working clearly.

A microchip manufacturer reduces chip size by a factor of 2 every 2 years (Moore's Law). If today's chip is 1 unit, what is the relative size after 10 years? Write as a power of 2.

A bacteria population is P = 3 x 2^t, where t is hours. How many bacteria are there after t = 0, t = 3, and t = 6 hours? Write in scientific notation.

Complex Simplification Challenge

Simplify each expression fully. Show all steps. Use multiple index laws in sequence.

(2x^3)^4 / (4x^5)^2 =

(a^2 b^-1)^3 x (a^-1 b^2)^2 =

3^(n+1) / 3^(n-1) =

TipThese challenging problems require students to apply multiple index laws in the correct order. Encourage your child to write each step separately and name the law being used.

Index Laws in Technology

Explore index laws in computer science and everyday technology.

1Research how computer memory uses powers of 2. Write each unit (kilobyte, megabyte, gigabyte, terabyte) as a power of 2 bytes. Use index laws to convert between them.
2Investigate how compound interest uses index laws: A = P(1 + r)^n. Use a calculator to find A if P = $1000, r = 0.05 (5%), and n = 10 years. How does this relate to a^n?
3Find a Rubik's Cube and research the number of possible combinations (approximately 4.3 x 10^19). Write this number as a power of 10 and use the quotient law to find how many more combinations there are than, say, 4.3 x 10^14.

Deriving the Quotient Law from First Principles

Show how the quotient law follows from the product law and negative indices.

Write a^m / a^n as a^m x a^___. Then apply the product law: a^m x a^-n = a^(m + ___) = a^(___). This derives the quotient law.

Use the same approach to simplify x^3 / x^7: write as x^3 x x^-7 = x^___ =

Index Laws and Graphs — Connecting Algebra to Geometry

Answer these questions about exponential function graphs.

For the function y = 2^x, complete the table: x = -2 gives y = ___, x = -1 gives y = ___, x = 0 gives y = ___, x = 1 gives y = ___, x = 2 gives y = ___.

In the table above, what happens to y as x increases by 1? This is related to which index law?

Write the equation of an exponential function that passes through (0, 3) and has a growth factor of 5 per unit.

TipGraphing exponential functions brings index laws to life visually. If you have access to graphing software, use it to plot the functions and check your child's predictions.

Fractional Exponents Preview

Use the pattern of exponents to answer these questions about fractional powers.

We know a^2 x a^2 = a^4. What should a^(1/2) x a^(1/2) equal? This means a^(1/2) = ___.

Using your answer above, evaluate: 9^(1/2) = ___. Check: square your answer.

Evaluate: 27^(1/3). (Hint: a^(1/3) is the cube root of a.) Check: cube your answer.

TipFractional exponents are a Year 10 topic, but a capable Year 9 student can discover them by extending patterns. These questions are an enrichment challenge, not a standard requirement.

Index Laws: Extending to (a/b)^n

Apply the power of a quotient rule: (a/b)^n = a^n / b^n.

(x/y)^4 =

(2x/3)^3 =

(a^2/b)^4 =

(3m^2 / n^3)^2 =

Index Laws in Algebra — Connecting to Expansion

These problems connect index laws to expanding algebraic expressions.

Expand: (x^2 + 1)(x^2 - 1). (Hint: difference of two squares)

Simplify: (x^3)^2 - (x^2)^3. Are they always equal?

Simplify: a^(m+1) / a^(m-1). Write as a simple power of a.

TipStudents who understand index laws as applying to any algebraic expression — not just standalone terms — are much better prepared for binomial expansion and factorising.

Comprehensive Index Laws Assessment

Show full working for all parts. This is the final index laws assessment.

Simplify: 5x^3 y^2 x 3x^2 y^4

Simplify: (2a^3 b^-2)^4

Simplify: 12m^8 n^5 / (4m^3 n^2)

Evaluate: (3^4 x 3^-2) / 3^0

Simplify and express with positive indices only: (x^-3 y^2)^2 / (x^2 y^-1)^3

TipThis comprehensive assessment covers all five index laws and their combinations. A student who can complete all five correctly is ready to move on to expanding binomial products.

Index Laws — Targeted Practice Set A

Simplify each expression. State which law you are using for each step.

p^7 x p^-4 =

q^3 / q^-2 =

(r^4)^-2 =

s^0 x s^3 =

(4t^2)^3 =

Index Laws — Targeted Practice Set B

Simplify each expression fully.

6a^4 b^3 x 2a^-2 b =

(x^3 y^-2)^3 =

15m^8 n^4 / (5m^5 n^2) =

(2x^2)^3 / (4x^3) =

TipSet B increases the difficulty slightly. Encourage your child to simplify step by step rather than trying to do everything at once.

Negative Index to Fraction Conversion

Write each expression without negative indices, using fractions.

a^-5 =

3b^-2 =

x^2 y^-3 =

2m^-1 n^4 =

(3a)^-2 =

4x^-2 y^-1 =

Evaluate with Negative and Zero Exponents

Circle the correct numerical value for each expression.

4^-1

1/4

-4

1/16

6^0

2^-5

1/32

-32

1/10

(-3)^-2

1/9

-1/9

Applying All Five Laws — Comprehensive Practice

Simplify each expression, applying the appropriate index laws at each step.

(a^3 b^2)^4 / (a^5 b^3)^2 =

(3x^-2)^2 x (2x^3)^3 / (6x^4) =

(m^2 n^-3)^2 x (m^-1 n^2)^4 =

Index Laws Applied to Perimeter and Area

Use index laws to answer these geometry questions.

A square has side length x^3 cm. Write expressions for: (a) perimeter = ___, (b) area = ___.

A rectangle has length 2x^4 cm and width 3x^2 cm. Simplify: (a) area = ___, (b) perimeter = ___.

A cube has side length a^2 m. Simplify: (a) surface area = ___, (b) volume = ___.

TipGeometry problems give index law practice a concrete context. Encourage your child to draw a diagram before writing any algebra.

Index Law Expressions — Advanced Match

Draw a line from each unsimplified expression to its simplified form.

(2x^3)^2 x x^4

3a^4 / a^7

(b^2 c^-1)^3

x^-2 y^4 / (x^-1 y^2)

(4p^3)^0

4x^10

3a^-3

b^6 c^-3

x^-1 y^2

Applying Index Laws to Number Theory

Use prime factorisation and index laws to solve each problem.

Write 144 as a product of prime factors using index notation. 144 = ___

Write 360 = 2^3 x 3^2 x 5. Simplify: 720 / 360 using prime factored forms.

If 2^a x 3^b = 72, find a and b by expressing 72 as a product of prime factors.

Simultaneous Index Equations

Solve for both unknowns in each system of index equations.

2^x x 2^y = 2^8 and 2^x / 2^y = 2^2. Find x and y.

a^m x a^n = a^9 and m = 2n. Find m and n.

TipThese simultaneous index equations challenge students to combine algebraic reasoning with index law knowledge. This is extending content for confident students.

Index Laws in Algebra — Writing Expressions

Write an algebraic expression using index laws for each description.

Write an expression for a^m multiplied by itself n times.

Write the simplest form of x^n / x^n for any non-zero x.

If f(x) = x^3 and g(x) = x^4, write a simplified expression for f(x) x g(x) and f(x) / g(x).

Write an expression for (2^n)^2 simplified, then for (2^n)^2 x 2^(1-n).

Exponent Laws Review: Fluency Drill

Simplify as quickly as you can. Time yourself and aim for under 10 minutes for all 10.

x^6 x x^3 = ___ x^8 / x^2 = ___ (y^4)^3 = ___ 5^0 = ___ z^-2 = ___ (3m^2)^2 = ___ a^5 x a^-7 = ___ b^3 / b^-4 = ___ (c^2 d)^3 = ___ 4p^3 x 2p^-1 = ___

TipFluency drills build automaticity — the ability to apply index laws without stopping to think. This is valuable for exam conditions where time is limited.

Index Laws with Irrational Bases

Apply index laws even when the base is irrational.

(sqrt(2))^4 = (sqrt(2))^4 = (2^(1/2))^4 = 2^___ =

(sqrt(3))^6 =

sqrt(2) x sqrt(2) x sqrt(2) = (sqrt(2))^___ = 2^___ =

pi^2 x pi^3 = pi^___, (can this be simplified further?)

TipIndex laws apply regardless of whether the base is rational or irrational. This connects the index laws topic to the surds topic covered earlier.

Final Review — All Five Index Laws

For each expression, state which law(s) you used and simplify fully.

a^3 x a^4 (Law: ___) =

b^7 / b^3 (Law: ___) =

(c^5)^2 (Law: ___) =

d^0 (Law: ___) =

e^-4 (Law: ___) =

Exponent Laws — Comprehensive Final Assessment

Show full working. This is the final assessment for exponent laws.

Simplify: (4x^3)^2 x (2x^-1)^3

Write with positive indices only: (a^-2 b^3) / (a^4 b^-1)

Evaluate: (3^2 x 3^-4)^2

Solve for x: 2^x x 2^3 = 2^7

A rectangle has length 3x^4 and width 2x^-1. Find its area and express with a positive index.

TipA score of 4 out of 5 correct on this assessment indicates readiness to move on to expanding binomial products.

Index Laws — Quick Simplification Drill 1

Simplify each expression in one step.

m^4 x m^5 = ___

n^10 / n^3 = ___

(p^3)^5 = ___

q^0 = ___

r^-6 = ___

(2s^3)^4 = ___

3t^2 x 4t^5 = ___

u^7 / u^-2 = ___

TipSpeed through these — the goal is fluency, not working shown.

Evaluating Integer Powers

Evaluate each numerical expression.

3^4 =

(-2)^5 =

(-3)^4 =

4^-3 =

(2/3)^4 =

(-1)^100 =

Even or Odd Exponent Effects

Circle the correct result.

(-5)^2

-25

(-4)^3

-64

(-1)^99

-1

(-2)^6

-64

-12

Index Laws and Area/Volume

A cube has side length 2a^2 cm. Use index laws to find expressions for its surface area and volume.

Side = 2a^2. Surface area = 6 x (2a^2)^2 = 6 x ___ =

Volume = (2a^2)^3 = 2^3 x a^___ =

If a = 3, evaluate both the surface area and volume.

TipSurface area of a cube = 6s^2. Volume = s^3. Substitute s = 2a^2 and apply index laws.

Simplify and Match — Set B

Draw a line from each expression to its simplified form.

(5m^2)^2

n^6 / n^9

(3p^-1)^3

4q^5 x q^-5

(r^4)^1/2 (assume positive r)

25m^4

n^-3

27p^-3

r^2

Connecting Powers to Scientific Notation

Use index laws to simplify, then convert to scientific notation.

(10^4)^2 x 10^-3 = 10^___ = (standard form)

(2 x 10^3)^2 x (3 x 10^-1) = ___ x 10^___ = (scientific notation)

10^7 / 10^9 = 10^___ = (standard decimal)

Index Equations

Find the value of n in each equation.

3^n = 81. n =

2^n = 1/8. n =

5^n = 25. n =

4^n = 1. n =

10^n = 0.001. n =

TipIndex equations reinforce understanding of what exponents mean. The strategy is to write both sides as powers of the same base.

Sort Laws by Complexity

Sort these index expressions from simplest (fewest laws needed) to most complex.

x^3 x x^4

(2x^2)^3 / x^4

a^0

(a^2 b^-1)^3 x (a^-1 b)^2

m^-3

(p^3)^2 x p^-5

1 law

2 laws

3+ laws

Explain It in Words

Explain each index law in your own words, then give an example.

The product law (a^m x a^n = a^(m+n)) works because... Example:

The zero index law (a^0 = 1) makes sense because... Example:

The negative index law (a^-n = 1/a^n) means... Example:

TipExplaining a mathematical rule in plain language is one of the best tests of genuine understanding.

Mixed Proof by Example

Verify each index law identity numerically. Use a = 3, m = 2, n = 4 unless specified.

a^m x a^n = a^(m+n): LHS = 3^2 x 3^4 = ___ x ___ = ___. RHS = 3^6 = ___. Equal? ___

a^m / a^n = a^(m-n): LHS = 3^4 / 3^2 = ___ / ___ = ___. RHS = 3^2 = ___. Equal? ___

(a^m)^n = a^(mn): Use a = 2, m = 3, n = 2. LHS = (2^3)^2 = ___^2 = ___. RHS = 2^6 = ___. Equal? ___

Advanced: Simplify Expressions Involving n

These expressions contain n as an unknown exponent. Simplify using index laws.

a^(n+2) x a^3 = a^(___)

b^(2n) / b^n = b^(___)

(c^n)^3 = c^(___)

d^(3n) x d^(-n) x d^(2-n) = d^(___)

Explore Exponential Functions Using Desmos

Use free graphing software (desmos.com) to investigate exponential functions.

1Graph y = 2^x, y = 3^x and y = 5^x on the same axes. Describe what changes when the base increases.
2Graph y = (1/2)^x. What happens? Compare it to y = 2^x. How are they related?
3Graph y = 2^x and y = 2^(-x). Write two observations about symmetry.

Index Laws — Final 5-Minute Challenge

Simplify all five expressions in under 5 minutes. Show any working needed.

(a) x^4 y^-3 x x^-2 y^5 =

(b) (3a^2 b^-1)^3 =

(d) (m^6 n^4)^(1/2) = m^___ n^___ (assume positive)

(e) Evaluate: (3^-2 x 3^5) / 3^2 =

Match Exponential and Radical Forms

Draw lines to match each exponential expression to its radical equivalent.

x^(1/2)

x^(1/3)

x^(3/2)

x^(2/3)

x^(-1/2)

sqrt(x)

cube_root(x)

sqrt(x^3) = x*sqrt(x)

(cube_root(x))^2

1/sqrt(x)

TipCover the right column and work out each exponential form step by step.

Circle the Larger Value

Circle the expression that has the greater value.

Which is larger?

2^10

10^2

Which is larger?

3^4

4^3

Which is larger?

(-2)^4

(-2)^3

Which is larger?

5^0

0^5

Proof Using Index Laws

Show algebraically that each statement is true. Write each step.

Prove that (a^2 b^3)^2 / (a^3 b^2) = a b^4

Prove that (x^3)^2 x x^0 = x^6

Show that 4^3 x 2^(-4) = 4 using index laws and base 2.

TipWrite out every step — marks are awarded for method, not just the answer.

Exponential Sequences

Continue each sequence and identify the common ratio.

TipFind what you multiply by each time — this is the base of the power.

Sort Expressions by Value

Place each expression in the correct column: Less than 1, Equal to 1, or Greater than 1.

2^(-3)

5^0

3^2

10^(-1)

(1/2)^(-1)

1^100

Less than 1

Equal to 1

Greater than 1

TipEvaluate each expression numerically first, then sort.

Compound Index Challenges

Simplify fully, writing each step.

Simplify: (2a^3 b^2)^2 x (3a b^(-1))^2

Simplify: (x^4 y^2)^3 / (x^2 y)^4

Simplify: [(m^2 n^(-3))^2 x m^(-1) n^4]^3

TipTackle one operation at a time: brackets first, then multiply/divide, then combine like bases.

Powers in the Real World

Explore exponential growth and decay with these family activities.

1Fold a piece of paper in half repeatedly and count the layers: after n folds, there are 2^n layers. Try to reach 7 folds.
2Look up the current Australian population and calculate how many times it has doubled since 1800 (was about 350,000).
3Research the half-life of carbon-14 (5,730 years). If you start with 1 kg, how much remains after 3 half-lives?
4Use a spreadsheet to plot y = 2^x for x = -4 to 4. Print or screenshot the graph and label key points.

Index Laws Mastery Self-Check

Rate your confidence for each law (1 = still confused, 5 = totally confident) and write an example of each.

Product law (a^m x a^n). Confidence: ___ / 5. My example:

Quotient law (a^m ÷ a^n). Confidence: ___ / 5. My example:

Power law ((a^m)^n). Confidence: ___ / 5. My example:

Zero exponent. Confidence: ___ / 5. My example:

Negative exponents. Confidence: ___ / 5. My example:

Fractional exponents. Confidence: ___ / 5. My example:

TipHonest self-assessment helps identify what to revise before a test.

Index Laws in Finance

Use index laws to solve these compound interest problems.

A = 1000 x (1.05)^3. Write out the expanded multiplication and find A.

If P = $2000, r = 0.04, n = 2, write A = P(1+r)^n with numbers and calculate.

Why is the exponent in compound interest equal to the number of years?

TipCompound interest formula: A = P(1 + r)^n — the exponent n is the number of periods.

Create Your Own Index Laws Problems

Design three challenging problems covering different index laws, then provide worked solutions.

Problem 1 (product and quotient laws):

Worked solution:

Problem 2 (power law and negative exponents):

Worked solution:

Problem 3 (fractional exponents):

Worked solution:

TipCreating problems builds the deepest understanding of a topic.

Correct Simplification

Circle the correctly simplified expression in each row.

a^6 / a^6 = ?

undefined

a^0 = 1

5^(-2) = ?

-25

-10

1/25

(xy)^3 = ?

x^3 y

xy^3

x^3 y^3

3xy

x^(3/2) in radical form = ?

x^3/2

sqrt(x^3)

3/2 * x

x^3 + x^(1/2)

100

Unit Review: Index Laws

Exam-style review. Set a 15-minute timer and work without notes.

State the product law, quotient law and power law in words and symbols.

Simplify fully: (2x^3 y^(-2))^2 x (x y^3) / (4 x^5 y^(-1))

Evaluate: 125^(2/3) - 4^(3/2) + 3^0

A bacteria colony triples every hour. Write the count after t hours as a power. How many after 8 hours starting from 50?

TipRealistic exam conditions are the best preparation — avoid the temptation to check notes.

What to do next

Algebra

Expanding Binomial Products

Use the distributive law (FOIL) to expand products of two binomial expressions, including difference of two squares and perfect squares.