Exponent laws: a week of ready-to-teach maths

Five lessons for Year 8 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus factor tiles to cut out, power match cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair, and cut them before Day 2: the factor tiles are for Days 2 to 4 and the power match cards for Day 4.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

An exponent counts equal factors: 2³ means 2 × 2 × 2, three 2s, which is 8. Every exponent law is just careful counting of those factors. Multiplying 2³ by 2⁴ joins three 2s to four 2s, so there are seven: the exponents add. Dividing 2⁵ by 2² cancels two pairs and leaves three 2s: the exponents subtract. Squaring 2³ writes the group of three 2s twice, six 2s in all: the exponents multiply. Stepping down from 2³ = 8 by dividing by 2 each time gives 4, 2 and then 1, so 2⁰ = 1. The laws only work when the bases are the same, and students should be able to check any of them by working out both sides.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write powers

The exponent is the small raised number and the base is the number being multiplied: in 5³, 5 is the base and 3 the exponent. Read 5³ as “5 to the power 3” or “5 cubed”. A number with no exponent has the exponent 1, so 10 = 10¹. We write powers with raised numbers, never with a caret. This unit keeps to whole-number exponents and the zero exponent, with numbers only: negative exponents and letters come later. Calculators are for checking, after the law.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 makes sure an exponent is read as a count of factors, not a multiplier; Days 2, 3 and 4 build the three laws from that count, each with factor tiles before any rule is written; and Day 5 extends the pattern down to the zero exponent and puts all the laws to work. It builds on Year 7, where students met square numbers, square roots and powers of primes. It prepares scientific notation, index laws with letters in Year 9 and the growth and decay models of later years.

Materials for the week (one trip)

• From the classroom: scissors, pencils, an envelope per pair for the tiles, this pack printed.
• Cut-out sheet 1, one per pair: 24 factor tiles of 2 and 12 of 3, with a fraction bar strip (Days 2 to 4).
• Cut-out sheet 2, one per pair: 24 power match cards (Day 4).
• Calculators for checking on Days 1, 2 and 5. A few sheets of scrap paper for the Day 1 folding stretch. No other equipment to buy.

Words for the word wall

base, exponent, power, index, index form, expanded form, squared, cubed, factor, product, quotient, product law, quotient law, power of a power, zero exponent, simplify, evaluate, cancel, same base

Dear families

This week in maths, Year 8 is learning the exponent laws. An exponent counts how many times a number is multiplied by itself: 2⁵ means 2 × 2 × 2 × 2 × 2, which is 32. The laws are short cuts for counting those factors: multiply powers of the same number and the exponents add, divide and they subtract, raise a power to a power and they multiply. Any number (except 0) to the power 0 is 1.

Try this at home

• Fold a sheet of paper in half again and again, counting the layers each time: 2, 4, 8, 16... How many folds before you cannot fold any more?
• Ask: is 2³ the same as 2 × 3? (No: 8 against 6.)
• Ask: what is 10² × 10³? (10⁵, which is 100 000: two zeros and three zeros make five.)
• Ask: what is 5⁰? (1, like any number other than 0 to the power 0.)

My powers this week

Fill one row a day. Write a power, its expanded form and its value. Use a different base each day.

Printed from the free seegongsik Exponent laws teaching pack · seegongsik.com/au/y8/number/AC9M8N02/pack

An exponent as repeated multiplication

An exponent counts equal factors. Students move between index form, expanded form and value, and compare pairs such as 2⁵ and 5² to see that swapping base and exponent changes the number.

We are learning to

• name the base and the exponent of a power and write it in expanded form,
• write a product of equal factors in index form and evaluate a power,
• explain why 2³ is 8 and not 6, and why 2⁵ and 5² are different numbers.

Success criteria

• I can write 3 × 3 × 3 × 3 as 3⁴ and work out its value.
• I can say which is bigger, 3² or 2³, and show the working.

You need

The worksheet, one per student; calculators for checking only; scrap paper for the Part D folding.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the board again.

Watch for these ideas

• Multiplying base by exponent: 2³ is 2 × 2 × 2 = 8, not 2 × 3 = 6.
• Thinking the order does not matter: 2⁵ = 32 but 5² = 25.
• Writing 10⁵ with five zeros but forgetting the 1, or with the wrong number of zeros.

Answers

• Part A: 3⁴ = 3 × 3 × 3 × 3 = 81; 5³ = 5 × 5 × 5 = 125; 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64; 10⁵ = 10 × 10 × 10 × 10 × 10 = 100 000; 1⁷ = 1 × 1 × 1 × 1 × 1 × 1 × 1 = 1; 7² = 7 × 7 = 49.
• Part B: 6⁴; 10³; 2³ × 3²; 4⁵; 7.
• Part C: 2⁵ = 32 > 5² = 25; 3⁴ = 81 > 4³ = 64; 2¹⁰ = 1024 > 10² = 100; 2⁴ = 16 = 4² = 16.
• Part D: after n folds there are 2ⁿ layers; 2⁹ = 512 is under 1000 and 2¹⁰ = 1024 is over, so 10 folds. At 0.1 mm a sheet the stack is 102.4 mm thick, about 10 cm.

An exponent as repeated multiplication

A. Expanded form and value

B. Index form

Write each product in index form. Group equal factors first.

C. Which is bigger?

Work out both values, then write >, < or = between the powers.

D. Stretch: folding paper

Each fold doubles the number of layers. Write the layers after 1, 2, 3, 4 and 5 folds as powers of 2. How many folds would give more than 1000 layers? If one sheet is 0.1 mm thick, how thick is that stack?

The product law

Multiplying powers of the same base joins their factors, so the exponents add. Students build products with factor tiles, check the law by working out both sides, and find where it does not apply.

We are learning to

• explain with factors why a^m × aⁿ = a^m+n when the bases are the same,
• simplify a product of powers of the same base to a single power,
• check the law by working out both sides, and say when it cannot be used.

Success criteria

• I can simplify 3² × 3⁴ to 3⁶ and show it with factors.
• I can explain why 2³ × 3² cannot be written as one power.

You need

Factor tiles (cut-out sheet 1), one envelope per pair; the worksheet; calculators for Part B.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the tiles; start the second with Part A.

Watch for these ideas

• Multiplying the exponents: 2³ × 2³ is 2⁶, not 2⁹.
• Multiplying the bases: 3² × 3² is 3⁴ = 81, not 9⁴.
• Forgetting the hidden exponent: 10 × 10⁴ = 10⁵, because 10 = 10¹.

Answers

• Part A: 2³ × 2⁴ = 2⁷; 5 × 5⁶ = 5⁷; 10² × 10³ × 10 = 10⁶; 3⁴ × 3⁴ = 3⁸; 7² × 7⁵ = 7⁷.
• Part B: 4 × 8 = 32 = 2⁵; 3 × 9 = 27 = 3³; 100 × 100 = 10 000 = 10⁴.
• Part C: 1 true (the exponents add: 2 + 3 = 5); 2 false (different bases: 8 × 9 = 72, but 6⁵ = 7776); 3 false (it is 2⁶: the exponents add, they do not multiply); 4 true (10 is 10¹, and 4 + 1 = 5); 5 false (it is 3⁴ = 81; 9⁴ = 6561); 6 true (1 + 2 + 3 = 6).
• Part D: 1. 7; 2. 4; 3. 6; 4. 5 (8 = 2³ and 27 = 3³).

The product law

A. One power

Write each product as a single power.

B. Check it both ways

Work out each power, multiply, and compare with the single power the law gives.

C. True or false?

Circle T or F. For each false line, write a value that proves it.

D. Stretch: the missing exponent

Find the number that goes in each box. For the last two, first write 8 and 27 as powers.

The quotient law

Dividing powers of the same base cancels matching factors, so the exponents subtract. Students cancel factor tiles across a bar, then simplify products and quotients in one line.

We are learning to

• explain with cancelled factors why a^m ÷ aⁿ = a^m−n for the same base,
• simplify a quotient of powers to a single power,
• simplify expressions that mix products and quotients, working from the brackets out.

Success criteria

• I can cancel factors to show that 2⁶ ÷ 2⁴ = 2².
• I can simplify a line such as (3⁴ × 3⁵) ÷ 3⁶ and check it by working out the values.

You need

Factor tiles and the fraction bar strip (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part A.

Watch for these ideas

• Dividing the exponents: 2⁶ ÷ 2² is 2⁴, not 2³.
• Dividing the bases as well: 10⁶ ÷ 10² is 10⁴, not 1⁴.
• Losing the last factor: 4⁵ ÷ 4 = 4⁴, because 4 = 4¹.

Answers

• Part A: 2⁷ ÷ 2³ = 2⁴; 10⁶ ÷ 10² = 10⁴; 5⁹ ÷ 5⁸ = 5¹ = 5; 3¹⁰ ÷ 3⁵ = 3⁵; 4⁵ ÷ 4 = 4⁴.
• Part B: 3⁶ ÷ 3²: 6 3s over 2, 2 pairs cancel, 4 left = 3⁴ = 81; 10⁵ ÷ 10³: 5 10s over 3, 3 pairs cancel, 2 left = 10² = 100.
• Part C: (3⁵ × 3²) ÷ 3⁴ = 3⁷ ÷ 3⁴ = 3³; (10⁴ × 10³) ÷ 10⁵ = 10⁷ ÷ 10⁵ = 10²; (2⁹ ÷ 2³) × 2 = 2⁶ × 2 = 2⁷; 5⁸ ÷ (5² × 5³) = 5⁸ ÷ 5⁵ = 5³.
• Part D: by the law 2⁴ ÷ 2⁴ = 2⁰; but any number divided by itself is 1 (16 ÷ 16 = 1). Both are right, so 2⁰ must be 1.

The quotient law

A. One power

Write each quotient as a single power.

B. Write the factors and cancel

Write every factor above and below a bar, cross out matching pairs, and write what is left as a power and a number.

C. Products and quotients together

Do the brackets first. Write the middle step.

D. Stretch: when top and bottom match

Work out 2⁴ ÷ 2⁴ two ways: with the quotient law, and by dividing the value of 2⁴ by itself. What must 2⁰ be? Does the same argument work for 5⁰?

A power raised to a power

Raising a power to a power writes the same group of factors again and again, so the exponents multiply. Students copy groups of factor tiles, tell this law from the product law, and match expressions to powers.

We are learning to

• explain with groups of factors why (a^m)ⁿ = a^m×n,
• decide whether a line needs the product law or the power-of-a-power law,
• write a number such as 8² as a single power of a prime.

Success criteria

• I can show with groups that (10²)³ = 10⁶ = 1 000 000.
• I can explain why (2³)² and 2³ × 2² give different powers.

You need

Factor tiles (cut-out sheet 1) and the power match cards (cut-out sheet 2), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the match cards; start the second with Part A.

Watch for these ideas

• Adding instead of multiplying: (5²)⁴ is 5⁸, not 5⁶.
• Raising the base: (10³)³ is 10⁹, not 1000³ worked out the long way.
• Thinking (2³)² = 2³ × 2²: the second line has only five factors.

Answers

• Part A: (2³)² = 2⁶; (5²)⁴ = 5⁸; (10³)³ = 10⁹; (7⁴)¹ = 7⁴; (3¹)⁵ = 3⁵.
• Part B: (2³)² = 2⁶ but 2³ × 2² = 2⁵; (3²)³ = 3⁶ but 3² × 3³ = 3⁵; (10⁴)² = 10⁸ but 10⁴ × 10² = 10⁶.
• Part C: 4³ = (2²)³ = 2⁶; 9² = (3²)² = 3⁴; 25³ = (5²)³ = 5⁶; 16² = (2⁴)² = 2⁸.
• Part D: (2³)² = 2⁶ = 64, but 2 to the power 3² is 2⁹ = 512. The brackets decide which power is taken first, so they cannot be left out.

A power raised to a power

A. One power

B. Power of a power, or product?

Simplify both lines in each row. Then say in a few words why they differ.

C. Powers of a prime

Write the number inside as a power of a prime first, then use the law. Example: 8² = (2³)² = 2⁶.

D. Stretch: where do the brackets go?

Work out (2³)². Then work out 2 to the power 3², squaring the 3 first. Are they the same? What job do the brackets do?

The zero exponent

Stepping down a ladder of powers divides by the base each time, so the step below a¹ is a⁰ = 1. Students use the zero exponent with all three laws, fix common mistakes and play the quiz as a class game.

We are learning to

• explain from a pattern, and from the quotient law, why a⁰ = 1 for any base other than 0,
• simplify expressions that need more than one law, including zero exponents,
• find and correct mistakes in exponent working.

Success criteria

• I can explain why 4⁰ = 1 in two different ways.
• I can find the mistake in “2⁶ ÷ 2³ = 2²” and correct it.

You need

The worksheet, one per student; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Thinking a⁰ = 0: the ladder and 2³ ÷ 2³ = 1 both say 1.
• Thinking a⁰ = a: 7⁰ is 1, while 7¹ is 7.
• Using one law for everything: count the factors when unsure.

Answers

• Part A: 5⁰ = 1; 10⁰ × 10² = 100; 1⁰ = 1; 2⁰ + 2³ = 9; 7¹ − 7⁰ = 6.
• Part B: (2⁴ × 2³) ÷ 2⁵ = 2⁷ ÷ 2⁵ = 2² = 4; (3²)³ ÷ 3⁴ = 3⁶ ÷ 3⁴ = 3² = 9; 10⁵ × 10⁰ = 10⁵ × 1 = 10⁵ = 100 000; (5³ × 5³) ÷ (5²)³ = 5⁶ ÷ 5⁶ = 5⁰ = 1.
• Part C: 1. 3⁶: keep the base, add the exponents; 2. 2⁶: subtract the exponents, do not divide them; 3. right: 125 × 8 = 1000 (equal exponents, so the bases multiply); 4. 4⁶: a power of a power multiplies the exponents; 5. 6⁰ = 1.
• Part D: 2³ × 2⁰ = 2³ by the product law, so 8 × 2⁰ = 8. If 2⁰ were 0 the left side would be 0, so 2⁰ must be 1. For 0⁰ the ladder would divide by 0, so it is left undefined.

The zero exponent

A. Work out

B. Simplify, then work out

Write the middle step, the single power and its value.

C. Spot the mistake

Four of these lines are wrong and one is right. Tick the right one; correct the others.

D. Stretch: what if 2⁰ were 0?

Use the product law on 2³ × 2⁰. What would happen to the answer if 2⁰ were 0? Why do mathematicians leave 0⁰ undefined? (Think about the ladder for base 0.)

Factor tiles and a fraction bar

For Days 2 to 4, one set per pair (keep them in an envelope)

Each tile is one factor. Lay tiles in a row to build a power, push two rows together to multiply, put one row above the bar and one below to divide, and lay out copies of a group to raise a power to a power.

Fraction bar (cut along the dashes)

Teacher note: 24 tiles of 2 and 12 tiles of 3 cover every tile task in the week; the largest is 2⁶ ÷ 2⁴ (ten 2-tiles).

Power match cards

For Day 4, one set per pair: twelve expressions

Twelve single powers

Teacher note: the matches are 2³ × 2⁴ → 2⁷; 3⁶ ÷ 3² → 3⁴; (5²)³ → 5⁶; 10⁴ × 10 → 10⁵; 2¹⁰ ÷ 2⁷ → 2³; (3³)² → 3⁶; 5⁴ × 5⁰ → 5⁴; (10²)² → 10⁴; 2² × 2² × 2² → 2⁶; 3⁵ × 3³ ÷ 3⁷ → 3; 7⁴ ÷ 7⁴ → 1; 5⁹ ÷ 5⁷ → 5².

What we know: Exponent laws

Work on your own, without a calculator. Show your working.

1. Write 3 × 3 × 3 × 3 × 3 in index form and work out its value. ______________________
2. Work out 2⁶ and 6². Which is bigger? ______________________
3. Simplify 7³ × 7⁵ to a single power. ______________________
4. Simplify 10⁹ ÷ 10⁴ and write the answer as an ordinary number. ______________________
5. Simplify (2⁴)³. ______________________
6. Work out 6⁰ + 2³. ______________________
7. Simplify (5⁶ × 5²) ÷ 5⁵, then work out its value. ______________________
8. Mia writes 2³ × 2⁴ = 4⁷. Explain the mistake and correct it. ______________________

Answers and marking guide

Answers with working

1. 3⁵ = 243.
2. 2⁶ = 64 and 6² = 36, so 2⁶ is bigger.
3. 7⁸: the exponents add, 3 + 5 = 8.
4. 10⁵ = 100 000.
5. 2¹²: the exponents multiply, 4 × 3 = 12.
6. 1 + 8 = 9.
7. 5⁸ ÷ 5⁵ = 5³ = 125.
8. The bases do not multiply: keep the base and add the exponents, 2³ × 2⁴ = 2⁷ = 128(Mia’s 4⁷ is 16 384).

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and can check a law by working out both sides.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: powers as factors, the product law, the quotient law, a power of a power and the zero exponent.