Simplify one step at a time: powers of powers first, then multiply and divide from left to right, keeping each base. The laws only join powers of the same base: 2³ × 3² cannot become one power. They also make scientific notation quick.
Scientists write the distance to the Sun as 1.5 × 10⁸ km. Multiplying numbers like that takes seconds with the index laws, as long as you only join powers of the same base.
Simplify an expression with several powers one law at a time, keeping track of each base:
(23)2×24÷25=26×24÷25=210÷25=25
The laws only work for powers of the same base. 23×32=8×9=72: it is not 65, and it cannot be written as one power of a whole number by the laws.
In scientific notation, the powers of 10 follow the laws: (3×104)×(2×103)=6×107.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Different bases
TryTry it on the board and compare the result with your prediction.
Many ways to 3⁸
Try it yourselfYour edits are saved in this browser.
Make 3^m × 3^n equal to 3⁸. Can you find another way?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
One step at a time
Here is the routine.
Worked exampleBefore each step, say what you think comes next.
(23)2×24÷25Powers of powers first.
=26×24÷253 × 2 = 6.
=210÷25Then left to right: 6 + 4 = 10.
=25=3210 − 5 = 5.
Read Lia's working. Are the bases the same?
Simplify 5⁴ × 5³ ÷ 5⁵. 5 to what power?
Lia simplifies 2⁴ × 5². Click the line where her working goes wrong.
ModifyChange the problem on purpose and see what happens.
Combinations
Need a hint?
64 = 2⁶.
Another hint
Which m and n multiply to 6?
Try it yourselfYour edits are saved in this browser.
Simplify (2⁴)³ ÷ 2¹⁰. 2 to what power?
Make a power of a power of 2 that equals 64.
Use the board. This ticks when you get there.
Make a division of powers of 10 that equals 1000.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Using the laws
Simplify (3²)³ × 3. 3 to what power?
Simplify 6⁸ ÷ 6³ ÷ 6². 6 to what power?
Review: what is 4³?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Build an expression
Then write an expression using all three laws that simplifies to 2⁵, and swap with a partner to check.
Try it yourselfYour edits are saved in this browser.
Need a hint?
256 = 2⁸.
Another hint
Which m and n multiply to 8?
Show a power of a power of 2 whose value is 256 on the board.
Use the board. This ticks when you get there.
For teachers
Students combine the index laws one step at a time, check that the bases match before joining powers, and use the laws with powers of 10 (AC9M8N02, node 2). About 35 minutes.
I can simplify an expression using more than one index law.
I can tell when powers can be joined (same base) and when they cannot.
I can multiply numbers in scientific notation using the laws.
Before class
Do the Run board yourself first. Students need the index laws (previous lesson).
The one idea: apply one law at a time (powers of powers, then multiply and divide left to right), and only for powers of the same base. The common mistake is joining different bases (Lia's mistake).
Adding powers (2³ + 2³) is not covered by the laws: a good discussion.
Wider: dividing in scientific notation; a systematic order for long expressions.
Running the lesson
Predict: ask a '6⁷' student to work out 2⁵ × 3² as numbers.
Run: ask what all the ways have in common.
Investigate: ask Lia to check with numbers.
Modify: ask why 7³ + 7⁴ is not 7⁷.
Make: ask a partner to simplify their expression.
The hard part (on purpose)
Checking the bases first. Ask students to circle each base before they use any law.
When students get stuck
A student adds indices for different bases. Ask: Circle the bases: are they the same? Fix: Only the same base combines.
A student adds when there is a power of a power. Ask: How many groups, of how many factors? Fix: Powers of powers multiply the indices.
A student treats 3 as 3⁰. Ask: How many 3s is 3? Fix: 3 = 3¹.
Stretch and support
Stretch: Multiply (4 × 10⁵) × (5 × 10³) and give the answer in scientific notation (careful: 4 × 5 = 20).
Support: Write each expression out as factors for the first two problems.
Where it leads
This completes AC9M8N02.
Where it leads: scientific notation in science, negative indices and algebraic powers (Year 9).
What each step is for
different bases: Committing to whether 2⁵ × 3² can be one power shows whether they check the base before using a law.
many ways to 3⁸: Finding several products that make the same power shows the index sum is what matters.
one step at a time: Combining laws in order; different bases do not combine (Lia adds anyway); powers of 10 in scientific notation follow the laws.
combinations: Longer combinations, a power of a power to a given value, and choosing which expressions the laws can join.
using the laws: Combinations, a sum that does not combine, and a review evaluation, mixed; joining different bases is the planted slip.
build an expression: Writing an expression that simplifies to a target uses the laws backwards and in combination.
Now you can
Simplify an expression using more than one index law.
Tell when powers can be joined (same base) and when they cannot.
Multiply numbers in scientific notation using the laws.