CLASS 8 MATHS · PART 1 · CHAPTER 2 · STUDY NOTES
Power Play
Understand power play through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
UNDERSTAND · APPLY · CHECK
Doubling makes bigger changes each time
Adding and multiplying are different growth rules. Watch how their totals separate.Add the same amount or multiply by the same factor?
Total (tokens)
Step (whole step)
- Add 3 each step · Start at 1, then repeatedly add 3.
- Double each step · Start at 1, then repeatedly multiply by 2.
The same linear axes compare two fictional token rules. At step 0 both totals are 1. Straight segments join whole-step observations.
Read the plotted values
| Step | Total |
|---|---|
| 0 | 1 |
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
| 4 | 13 |
| 5 | 16 |
| 6 | 19 |
| Step | Total |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
- Linear growth
- Adding the same amount at every equal step.
- Exponential growth
- Multiplying by the same positive factor at every equal step.
A doubling rule starting with 1 gives 1, 2, 4, 8, 16, 32 and 64. The differences get larger, even though the multiplier stays 2.
Paper folding is a useful mathematical model of doubling thickness. Many folds are physically impossible for ordinary paper: the model is not a building plan.
Five doublings of an initial amount v
Final amount = v × 2⁵ = 32vKeep the starting amount. The factor 32 alone is not the final quantity unless v = 1.
Take a moment. What do you think?
PAUSE & TRY · think first, then check with a free account
Question
A model doubles daily and fills a container on day 12. On which day is it half full?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 1, 2, 3, 17, 18.
UNDERSTAND · APPLY · CHECK
The base is the repeated factor; the exponent counts it
Read the small raised number as an instruction to multiply equal factors.Count factors, not additions
- Base
- The number or expression being repeatedly multiplied.
- Exponent
- For a positive integer exponent, the number of equal factors.
3⁴ = 3 × 3 × 3 × 3 = 81. It is different from 4 × 3 = 12.
Parentheses matter. (−3)² = 9, while −3² means −(3²) = −9. Even powers of a negative base are positive; odd powers are negative.
Let’s look a little closer.
Go deeper: Zero as a base
For a positive integer n, 0ⁿ = 0. This chapter does not define 0⁰ or negative powers of zero.
PAUSE & TRY · think first, then check with a free account
Question
Write a × a × b × b × b in compact form.
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 3, 4, 5.
UNDERSTAND · APPLY · CHECK
Join same-base factor groups by adding exponents
When the bases match, join their lists of equal factors.Three factors join two factors
- Same-base product law
- aᵐ × aⁿ = aᵐ⁺ⁿ. For integer exponents that may be zero or negative, require a ≠ 0.
2³ × 2² has three factors of 2 followed by two more. Together they make five: 2⁵ = 32.
This rule is about multiplication. 2³ + 2² = 8 + 4 = 12; adding the exponents would give the wrong answer.
- 1
Check that both bases are 2.
- 2
Count 3 + 2 equal factors.
- 3
Write 2⁵ and, if needed, evaluate it as 32.
PAUSE & TRY · think first, then check with a free account
Question
Simplify 7² × 7⁴ without calculating a large integer.
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF page 6.
UNDERSTAND · APPLY · CHECK
A power of a power multiplies the counts
Expand the outer power as repeated groups, then count how many factors are inside all the groups.Three groups, two factors in each
- Power-of-a-power law
- (aᵐ)ⁿ = aᵐⁿ; for arbitrary integer exponents here, use a ≠ 0.
- Power of a product
- (ab)ⁿ = aⁿbⁿ, where the expressions are defined.
(2²)³ is three groups of two factors. It equals 2⁶ = 64.
A shared exponent lets you combine different bases: 2⁴ × 3⁴ = (2 × 3)⁴ = 6⁴. A power of a sum does not work this way.
A quotient with a shared exponent
12³ ÷ 4³ = (12/4)³ = 3³ = 27The denominator must be nonzero. This follows by dividing matching factors.
PAUSE & TRY · think first, then check with a free account
Question
Is (2 + 3)² equal to 2² + 3²?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 6, 7.
UNDERSTAND · APPLY · CHECK
Multiply choices when every pairing is allowed
Draw a small branching tree before working with a long code.Trace one branch for each permitted badge
- Multiplication principle
- If each of m choices can be followed by each of n choices, there are mn ordered combinations.
Suppose a badge has two colour choices and three symbol choices. Each colour can pair with every symbol, so there are 2 × 3 = 6 badges.
A four-slot code with digits 0–9 has 10⁴ possibilities only if repetitions and a leading zero are allowed. A four-digit number has different rules.
PAUSE & TRY · think first, then check with a free account
Question
How many three-slot codes use only A or B, allowing repetition?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 8, 9.
UNDERSTAND · APPLY · CHECK
Divide matching factors; zero power becomes 1
Cancellation explains subtraction of exponents.Cancel matching factors
- Quotient law
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ, for a ≠ 0 and integer m, n.
- Zero exponent
- a⁰ = 1 for a ≠ 0.
5⁴ ÷ 5² cancels two factors in the numerator and denominator. Two remain, giving 5².
Dividing any nonzero power by itself gives 1. The quotient law then gives aⁿ ÷ aⁿ = a⁰ = 1. Zero is excluded because division by zero is undefined.
PAUSE & TRY · think first, then check with a free account
Question
What is 13⁶ ÷ 13⁶? Explain using both cancellation and exponents.
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 9, 10.
UNDERSTAND · APPLY · CHECK
A negative exponent means a reciprocal
Keep dividing by the base as you move down one exponent at a time.Divide by 3 at every arrow
- Negative exponent
- a⁻ⁿ = 1/aⁿ, for nonzero a and positive integer n.
- Reciprocal
- The number that multiplies a nonzero number to give 1.
The sequence 3², 3¹, 3⁰, 3⁻¹, 3⁻² has values 9, 3, 1, 1/3 and 1/9. Each step divides by 3.
A negative exponent does not make the value negative. For example, 4⁻² = 1/16 is positive. A negative base and a negative exponent are different ideas.
Combine integer exponents
6⁻² × 6⁵ = 6³The same-base product law still applies: −2 + 5 = 3. The base is nonzero.
PAUSE & TRY · think first, then check with a free account
Question
Calculate (−2)⁻³.
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 10, 11, 12.
UNDERSTAND · APPLY · CHECK
Powers of ten explain both sides of the decimal point
Each step right in a decimal place-value chart divides the place value by 10.Follow the descending place values
- Place value
- The value contributed by a digit because of its position.
708.04 = 7 × 10² + 0 × 10¹ + 8 × 10⁰ + 0 × 10⁻¹ + 4 × 10⁻². Zero holds empty places.
Multiplying a number by 10 moves each digit to a place worth ten times as much. Describe the value change, rather than treating the decimal point as an object that moves.
PAUSE & TRY · think first, then check with a free account
Question
What does the digit 6 contribute in 25.063?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF page 12.
UNDERSTAND · APPLY · CHECK
Write a large or small number as a coefficient times 10
For a positive quantity, put one nonzero digit before the decimal point and record the scale with a power of 10.Keep the value unchanged
- Scientific notation
- For a positive number, c × 10ᵏ with 1 ≤ c < 10 and integer k.
- Coefficient
- The factor c that states the leading size within a power-of-ten scale.
47,200 = 4.72 × 10⁴. Since 10⁴ = 10,000, multiply to check the original number.
0.00062 = 6.2 × 10⁻⁴. The negative exponent records a small positive quantity. A lakh is 10⁵, a crore is 10⁷, a million is 10⁶ and a billion is 10⁹.
Go deeper: Zero and negative numbers
Zero cannot have a nonzero coefficient between 1 and 10. For a negative nonzero number, keep the minus sign and apply the same coefficient rule to its absolute value. A googol is 10¹⁰⁰; a googolplex is 10 raised to a googol, not 10 × a googol.
PAUSE & TRY · think first, then check with a free account
Question
Convert 3.06 × 10⁵ into an ordinary number.
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 13, 14, 18, 25.
UNDERSTAND · APPLY · CHECK
Compare scales and convert units before dividing
An exponent helps you see scale. For calculations, keep the units and coefficient as well.Use thousands as the common unit
Value (number)
- Number-line marks · Only horizontal position represents the number.
- 18 · (8, 0)
- 212 · (12, 0)
The two values are 8 thousand and 12 thousand. The coordinate scale is thousands of items, not individual items.
Read the plotted values
| Value | Height |
|---|---|
| 8 | 0 |
| 12 | 0 |
- Ratio of quantities
- One quantity divided by another expressed in the same units.
- Approximation
- A nearby value used with an explicit level of precision.
A fictional store has 6 × 10⁶ pencils and another has 3 × 10⁴. The first has (6/3) × 10² = 200 times as many.
Compare equal powers of ten by their coefficients. If exponents differ, first rewrite at the same scale. Rounded inputs lead to an approximate result, not an exact count.
Same units first
(2.4 × 10⁴ m) ÷ (6 × 10² m) = 40Read the quotient as (2.4 × 10⁴)/(6 × 10²). Metres cancel, leaving a dimensionless ratio.
PAUSE & TRY · think first, then check with a free account
Question
Which is greater, 8 × 10³ or 1.2 × 10⁴?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 19, 20, 21, 23, 24, 25.
UNDERSTAND · APPLY · CHECK
Convert seconds step by step, with stated assumptions
A power of ten may be a short expression but represent a very long time.Convert with the correct factors
- Conversion factor
- A ratio of equal amounts in different units, such as 60 seconds per minute.
There are 60 seconds in a minute, 3,600 in an hour and 86,400 in a day. Divide by the relevant conversion factor.
10⁶ seconds is about 11.57 days, whereas 10⁹ seconds is about 31.7 years using 365 days per year. A factor of 1,000 makes an enormous difference.
- 1
Choose the requested unit: days.
- 2
Use 1 day = 24 × 60 × 60 = 86,400 seconds.
- 3
Calculate 172,800/86,400 = 2.
Go deeper: Check claims with a model
For estimates, say what you assume. A fictional 180 km journey at 30 km per day takes 6 days under that daily-distance assumption. Rest days and changing speed would alter the answer. An estimate is useful without pretending to know exact real-world data.
PAUSE & TRY · think first, then check with a free account
Question
How many days are 1.728 × 10⁵ seconds?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 21, 22, 23, 24.